Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153082 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13304 |
| 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 | |
347 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 70 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)(5,30)(6,13)(7,37)(8,15)(9,18)(10,26)(11,34)(12,36)(14,32)(17,61)(19,59)(20,58)(21,70)(22,60)(24,75)(25,78)(27,48)(28,49)(29,47)(31,55)(33,89)(35,90)(38,50)(39,54)(40,101)(41,67)(42,93)(43,66)(44,68)(45,62)(46,107)(51,116)(52,95)(53,98)(56,96)(57,63)(64,120)(65,118)(69,115)(71,114)(72,76)(73,82)(74,80)(77,123)(79,122)(81,87)(83,124)(84,117)(85,99)(86,112)(88,97)(91,111)(92,105)(94,104)(100,108)(102,125)(103,119)(106,126)(109,128)(110,127)(113,121);(1,3)(2,9)(4,24)(5,31)(6,19)(7,38)(8,22)(10,45)(11,50)(12,41)(13,43)(14,55)(15,44)(16,30)(17,26)(18,34)(20,36)(21,32)(23,72)(25,79)(27,64)(28,65)(29,63)(33,92)(35,94)(37,42)(39,69)(40,47)(46,108)(48,111)(49,103)(51,114)(52,70)(53,115)(54,105)(56,104)(57,117)(58,88)(59,89)(60,90)(61,87)(62,107)(66,95)(67,116)(68,98)(71,96)(73,120)(74,122)(75,78)(76,82)(77,80)(81,91)(83,125)(84,126)(85,100)(86,121)(93,97)(99,106)(101,128)(102,112)(109,113)(110,123)(118,124)(119,127);(1,4)(2,10)(3,17)(5,25)(6,28)(7,27)(8,29)(9,40)(11,46)(12,47)(13,23)(14,48)(15,49)(16,57)(18,62)(19,24)(20,63)(21,64)(22,65)(26,36)(30,87)(31,91)(32,80)(33,83)(34,82)(35,84)(37,99)(38,100)(39,86)(41,103)(42,102)(43,45)(44,72)(50,113)(51,109)(52,73)(53,110)(54,74)(55,106)(56,112)(58,118)(59,61)(60,75)(66,76)(67,101)(68,119)(69,77)(70,108)(71,121)(78,89)(79,92)(81,97)(85,96)(88,117)(90,124)(93,126)(94,125)(95,127)(98,128)(104,122)(105,111)(107,116)(114,123)(115,120)Polycyclic presentation:
2^9|d^2=gh,e^2=i,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,c^b=cf,d^b=dg,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320516, #72320861, #72320903 |
| maximal normal subgroup of | #72320516, #72320861, #72320903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60278);