Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,13,22,36)(8,14,25,39)(9,27,30,41)(10,17,11,16)(15,28,42,29)(19,33,46,60)(20,48,50,62)(21,43,31,35)(23,52,47,37)(24,54,56,66)(26,44,32,40)(34,51,61,49)(38,58,65,57)(45,63,67,64)(53,59,55,68);(5,7)(6,13)(8,23)(9,20)(10,31)(11,21)(12,22)(14,37)(15,34)(16,43)(17,35)(18,36)(19,26)(24,55)(25,47)(27,48)(28,51)(29,49)(30,50)(32,46)(33,40)(38,64)(39,52)(41,62)(42,61)(44,60)(45,57)(53,56)(54,59)(58,67)(63,65)(66,68);(2,4)(5,8)(6,14)(7,19)(9,24)(10,32)(11,26)(12,25)(13,33)(15,38)(16,44)(17,40)(18,39)(20,45)(21,47)(22,46)(23,31)(27,54)(28,58)(29,57)(30,56)(34,59)(35,52)(36,60)(37,43)(41,66)(42,65)(48,63)(49,53)(50,67)(51,55)(61,68)(62,64);(5,9,12,30)(6,15,18,42)(7,20,22,50)(8,24,25,56)(10,29,11,28)(13,34,36,61)(14,38,39,65)(16,27,17,41)(19,45,46,67)(21,51,31,49)(23,53,47,55)(26,58,32,57)(33,59,60,68)(35,62,43,48)(37,63,52,64)(40,66,44,54)Polycyclic presentation:
2^9|a^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324821);