Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #452530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16051 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
872 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)(3,15,19,34)(4,20,25,38)(5,26,30,11)(6,12)(7,13)(8,14)(9,18,35,31)(10,24,39,23)(16,32)(17,47)(21,43)(22,53)(27,33)(28,42,29,41)(36,45)(37,55)(40,49)(44,61,64,58)(46,50)(48,59)(51,56)(52,60,63,62)(54,57);(1,3,7,18,8,19,6,31)(2,9,13,15,14,35,12,34)(4,21,24,16,25,45,23,46)(5,27,29,49,30,48,28,17)(10,36,20,50,39,43,38,32)(11,40,42,59,26,47,41,33)(22,44,55,63,56,64,54,52)(37,61,51,62,57,58,53,60);(1,4)(2,10)(3,16,19,46)(5,22)(6,24)(7,23)(8,25)(9,32,35,50)(11,37)(12,20)(13,38)(14,39)(15,43,34,36)(17,44,49,64)(18,21,31,45)(26,57)(27,52,48,63)(28,55)(29,54)(30,56)(33,58,59,61)(40,62,47,60)(41,51)(42,53);(1,5,8,30)(2,11,14,26)(3,17)(4,22,25,56)(6,28,7,29)(9,33)(10,37,39,57)(12,41,13,42)(15,40)(16,44)(18,27)(19,49)(20,51,38,53)(21,52)(23,54,24,55)(31,48)(32,58)(34,47)(35,59)(36,60)(43,62)(45,63)(46,64)(50,61)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359727);