Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24128 |
| 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 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,8,14,31)(4,9,19,36)(5,10,27,44)(6,11,25,42)(13,47,48,30)(15,32,24,41)(16,33,28,45)(17,51,52,34)(18,53,49,35)(20,37,55,61)(21,38,29,46)(22,39,57,63)(23,40,58,64)(26,43,56,62)(50,60,54,59);(1,3)(2,8)(4,17)(5,22)(6,28)(7,14)(9,34)(10,39)(11,45)(12,31)(13,20)(15,26)(16,25)(18,23)(19,52)(21,54)(24,56)(27,57)(29,50)(30,37)(32,43)(33,42)(35,40)(36,51)(38,60)(41,62)(44,63)(46,59)(47,61)(48,55)(49,58)(53,64);(1,4)(2,9)(3,13)(5,23)(6,29)(7,19)(8,30)(10,40)(11,46)(12,36)(14,48)(15,50)(16,18)(17,22)(20,26)(21,25)(24,54)(27,58)(28,49)(31,47)(32,59)(33,35)(34,39)(37,43)(38,42)(41,60)(44,64)(45,53)(51,63)(52,57)(55,56)(61,62);(1,3)(2,8)(4,18,19,49)(5,24)(6,16)(7,14)(9,35,36,53)(10,41)(11,33)(12,31)(13,29,48,21)(15,27)(17,23,52,58)(20,50,55,54)(22,56)(25,28)(26,57)(30,46,47,38)(32,44)(34,40,51,64)(37,59,61,60)(39,62)(42,45)(43,63)Polycyclic presentation:
2^9|a^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249759, #72249832, … |
| maximal normal subgroup of | #72249754, #72249759, #72249832, … |
| maximal quotient of | #72249934, #72250077, #72256856, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407670);