Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30593 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
533 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)(6,28)(7,12)(8,16)(9,20)(11,25)(13,26)(15,44)(17,37)(18,33)(21,55)(22,58)(23,31)(24,42)(27,38)(29,41)(30,56)(32,51)(34,46)(35,60)(36,57)(39,49)(40,52)(43,48)(45,54)(47,62)(50,61)(53,64)(59,63);(1,3,6,18,7,16,26,48)(2,8,13,33,12,14,28,43)(4,15,22,47,21,45,29,53)(5,23,25,51,27,17,49,52)(9,34,35,63,36,30,42,61)(10,37,39,32,38,31,11,40)(19,54,41,62,55,44,58,64)(20,56,24,59,57,46,60,50);(1,4)(2,9)(3,15,16,45)(5,24)(6,29)(7,21)(8,30,14,34)(10,19)(11,41)(12,36)(13,42)(17,50,23,59)(18,53,48,47)(20,25)(22,26)(27,60)(28,35)(31,54,37,44)(32,62,40,64)(33,63,43,61)(38,55)(39,58)(46,51,56,52)(49,57);(2,10)(3,16)(5,25)(6,26)(8,31)(9,19)(11,13)(12,38)(14,37)(15,45)(17,51)(20,24)(22,29)(23,52)(27,49)(28,39)(30,54)(32,33)(34,44)(35,58)(36,55)(40,43)(41,42)(46,59)(50,56)(57,60)(61,64)(62,63)Polycyclic presentation:
2^9|b^2=h,c^2=e,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,g^c=gi,h^c=hi,f^d=fh,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416540);