Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30400 |
| 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 | |
649 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)(4,19)(5,10)(6,29)(7,12)(8,16)(9,21)(11,26)(13,27)(15,45)(17,37)(18,34)(20,55)(22,58)(23,54)(24,32)(25,42)(28,38)(30,41)(31,56)(33,51)(35,57)(36,60)(39,49)(40,52)(43,46)(44,48)(47,62)(50,61)(53,64)(59,63);(1,3,6,18,7,16,27,48)(2,8,13,34,12,14,29,44)(4,20,22,53,23,15,30,47)(5,24,26,51,28,17,49,52)(9,31,36,61,35,43,42,63)(10,37,39,33,38,32,11,40)(19,45,41,64,54,55,58,62)(21,46,25,50,57,56,60,59);(1,4,7,23)(2,9,12,35)(3,15,16,20)(5,25,28,60)(6,30,27,22)(8,31,14,43)(10,19,38,54)(11,41,39,58)(13,42,29,36)(17,50,24,59)(18,53,48,47)(21,49,57,26)(32,55,37,45)(33,62,40,64)(34,63,44,61)(46,51,56,52);(2,10)(3,16)(5,26)(6,27)(8,32)(9,19)(11,13)(12,38)(14,37)(15,20)(17,51)(21,25)(22,30)(24,52)(28,49)(29,39)(31,55)(33,34)(35,54)(36,58)(40,44)(41,42)(43,45)(46,59)(50,56)(57,60)(61,64)(62,63)Polycyclic presentation:
2^9|b^2=h,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fh,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72320520, #72320582, #72320688 |
| maximal normal subgroup of | #72320520, #72320582, #72320688 |
| maximal quotient of | #72321310, #72321713, #72322148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416311);