Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23960 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20)(5,10)(6,27)(7,31)(8,16)(9,22)(11,30)(12,44)(13,26)(15,51)(17,45)(18,41)(19,36)(21,34)(23,46)(24,38)(25,47)(28,43)(29,49)(32,48)(33,42)(35,57)(37,54)(39,60)(40,59)(50,52)(53,62)(55,61)(56,63)(58,64);(1,3)(2,8)(4,21,24,52)(5,16)(6,28)(7,32)(9,37,40,51)(10,14)(11,41)(12,45)(13,48)(15,22,54,59)(17,30)(18,44)(19,26)(20,34,38,50)(23,55,33,56)(25,57)(27,35)(29,53,60,58)(31,36)(39,64,49,62)(42,63,46,61)(43,47);(1,4)(2,9)(3,15)(5,24)(6,29)(7,33)(8,34)(10,40)(11,42)(12,46)(13,49)(14,50)(16,54)(17,55)(18,56)(19,58)(20,27)(21,57)(22,30)(23,26)(25,60)(28,52)(31,39)(32,53)(35,62)(36,63)(37,41)(38,47)(43,64)(44,59)(45,51)(48,61);(1,3)(2,8)(4,15)(5,16)(6,18)(7,19)(9,34)(10,14)(11,43)(12,35)(13,36)(17,25)(20,51)(21,22)(23,53)(24,54)(26,32)(27,45)(28,44)(29,56)(30,57)(31,48)(33,58)(37,38)(39,61)(40,50)(41,47)(42,64)(46,62)(49,63)(52,59)(55,60)Polycyclic presentation:
2^9|b^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei,f^c=fh,g^c=gh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249759, #72249827, … |
| maximal normal subgroup of | #72249754, #72249759, #72249827, … |
| maximal quotient of | #72249931, #72250073, #72256853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407200);