Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13381 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2)(3,16)(4,23)(5,13)(6,33)(7,38)(8,15)(9,18)(10,26)(11,32)(12,36)(14,31)(17,62)(19,69)(20,44)(21,45)(22,43)(24,79)(25,81)(27,77)(28,88)(29,78)(30,91)(34,97)(35,57)(37,76)(39,53)(40,56)(41,80)(42,107)(46,113)(47,85)(48,101)(49,86)(50,82)(51,118)(52,120)(54,93)(55,84)(58,89)(59,64)(60,72)(61,71)(63,125)(65,121)(66,114)(67,110)(68,103)(70,124)(73,95)(74,122)(75,112)(83,106)(87,123)(90,119)(92,116)(94,96)(98,105)(99,109)(100,115)(102,117)(104,108)(111,127)(126,128);(1,3)(2,9)(4,24)(5,21)(6,34)(7,20)(8,22)(10,46)(11,43)(12,52)(13,16)(14,44)(15,45)(17,27)(18,32)(19,37)(23,62)(25,66)(26,83)(28,74)(29,63)(30,67)(31,71)(33,95)(35,100)(36,99)(38,103)(39,104)(40,75)(41,47)(42,55)(48,115)(49,59)(50,106)(51,60)(53,122)(54,109)(56,61)(57,105)(58,112)(64,82)(65,87)(68,89)(69,96)(70,101)(72,84)(73,94)(76,110)(77,125)(78,114)(79,81)(80,86)(85,113)(88,121)(90,126)(91,97)(92,98)(93,107)(102,127)(108,119)(111,116)(117,124)(118,120)(123,128);(1,4,8,29)(2,10,15,49)(3,17)(5,27,32,25)(6,35,37,102)(7,39,40,87)(9,41)(11,50,13,47)(12,53,55,123)(14,57,58,117)(16,59)(18,24)(19,70)(20,74)(21,63)(22,66)(23,76,78,33)(26,84,86,36)(28,89,90,31)(30,92,94,101)(34,98)(38,48,56,116)(42,108)(43,46)(44,111)(45,106)(51,119,54,88)(52,121)(60,122)(61,124)(62,69)(64,72)(65,71)(67,127)(68,104)(73,100)(75,126)(77,96,81,91)(79,110)(80,107)(82,93,85,118)(83,120)(95,125)(97,114)(99,113)(103,105)(109,128)(112,115)Polycyclic presentation:
2^9|c^2=dh,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320813, #72320882 |
| maximal normal subgroup of | #72320516, #72320813, #72320882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60459);