Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12731 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,2,6,13,8,12,36,15)(3,16,20,60,19,61,22,73)(4,23,25,55,28,10,27,46)(5,29,33,87,35,86,96,88)(7,37,32,14,39,56,95,54)(9,40,42,67,44,68,58,18)(11,48,52,97,51,111,53,34)(17,59,64,118,63,101,70,119)(21,72,71,113,69,115,38,49)(24,76,79,124,31,93,81,126)(26,83,82,47,85,98,78,75)(30,57,91,105,90,116,92,43)(41,100,102,62,74,99,104,66)(45,106,108,80,50,110,114,94)(65,121,84,112,120,128,122,107)(77,117,125,123,89,109,127,103);(1,3)(2,9)(4,17)(5,30)(6,20)(7,38)(8,19)(10,41)(11,49)(12,44)(13,42)(14,57)(15,58)(16,29)(18,34)(21,32)(22,36)(23,74)(24,77)(25,64)(26,84)(27,70)(28,63)(31,89)(33,91)(35,90)(37,43)(39,71)(40,48)(45,107)(46,102)(47,109)(50,112)(51,113)(52,72)(53,115)(54,116)(55,104)(56,105)(59,76)(60,87)(61,86)(62,80)(65,78)(66,94)(67,97)(68,111)(69,95)(73,88)(75,117)(79,125)(81,127)(82,120)(83,123)(85,122)(92,96)(93,101)(98,103)(99,110)(100,106)(108,121)(114,128)(118,124)(119,126);(1,4)(2,10)(3,17)(5,31)(6,27)(7,26)(8,28)(9,41)(11,50)(12,23)(13,55)(14,47)(15,46)(16,59)(18,66)(19,63)(20,70)(21,65)(22,64)(24,35)(25,36)(29,76)(30,89)(32,78)(33,79)(34,94)(37,98)(38,84)(39,85)(40,99)(42,104)(43,103)(44,74)(45,51)(48,110)(49,112)(52,108)(53,114)(54,75)(56,83)(57,109)(58,102)(60,119)(61,101)(62,67)(68,100)(69,120)(71,122)(72,121)(73,118)(77,90)(80,97)(81,96)(82,95)(86,93)(87,126)(88,124)(91,125)(92,127)(105,123)(106,111)(107,113)(115,128)(116,117)Polycyclic presentation:
2^9|a^2=f,d^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320367, #72320861, #72320873 |
| maximal normal subgroup of | #72320367, #72320861, #72320873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59364);