Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500001 |
| 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 |
| prime radical | #24288 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2932 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,19,41)(4,20,22,48)(5,26,31,12)(6,32,30,11)(7,34,29,13)(10,33,43,23)(15,51,52,58)(16,39,28,62)(17,38,55,47)(18,49,35,40)(21,59,60,57)(24,45,27,50)(25,44,36,46)(37,61,63,53)(42,54,64,56);(1,3)(2,9)(4,21,22,60)(5,17)(6,28)(7,35)(8,19)(10,42,43,64)(11,39)(12,47)(13,49)(14,41)(15,24,52,27)(16,30)(18,29)(20,57,48,59)(23,54,33,56)(25,53,36,61)(26,38)(31,55)(32,62)(34,40)(37,44,63,46)(45,51,50,58);(1,4)(2,10)(3,15)(5,27)(6,33)(7,36)(8,22)(9,37)(11,46)(12,48)(13,50)(14,43)(16,21)(17,53)(18,56)(19,52)(20,26)(23,30)(24,31)(25,29)(28,60)(32,44)(34,45)(35,54)(38,42)(39,51)(40,59)(41,63)(47,64)(49,57)(55,61)(58,62);(1,3)(2,9)(4,15)(5,28)(6,17)(7,18)(8,19)(10,37)(11,47)(12,39)(13,40)(14,41)(16,31)(20,58)(21,24)(22,52)(23,61)(25,54)(26,62)(27,60)(29,35)(30,55)(32,38)(33,53)(34,49)(36,56)(42,44)(43,63)(45,57)(46,64)(48,51)(50,59)Polycyclic presentation:
2^9|a^2=i,b^2=e,f^2=hi,g^2=h,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=gh,h^b=hi,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249757, #72249766, … |
| maximal normal subgroup of | #72249754, #72249757, #72249766, … |
| maximal quotient of | #72249912, #72249930, #72249932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407198);