Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453808 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,6,13,7,31)(3,14,19,42,17,9,18,34)(4,20,24,37,23,10,25,51)(5,26,30,39,28,40,29,11)(15,32,21,49,45,41,44,35)(16,46,48,33,27,58,47,38)(22,53,55,50,54,57,56,36)(43,63,64,60,59,61,52,62);(1,3,7,18,6,17,8,19)(2,9,12,34,13,14,31,42)(4,21,25,15,23,44,24,45)(5,27,29,48,28,16,30,47)(10,35,51,32,20,49,37,41)(11,38,26,46,39,33,40,58)(22,52,56,59,54,64,55,43)(36,61,53,62,50,63,57,60);(1,4,6,23)(2,10)(3,15)(5,22,28,54)(7,24,8,25)(9,32,14,41)(11,36)(12,37)(13,20)(16,43)(17,45)(18,21)(19,44)(26,57)(27,59)(29,55,30,56)(31,51)(33,60,38,62)(34,35,42,49)(39,50)(40,53)(46,61,58,63)(47,64)(48,52);(1,5,6,28)(2,11,13,39)(3,16,17,27)(4,22,23,54)(7,30,8,29)(9,33,14,38)(10,36,20,50)(12,40,31,26)(15,43,45,59)(18,48,19,47)(21,52,44,64)(24,56,25,55)(32,60,41,62)(34,46,42,58)(35,61,49,63)(37,53,51,57)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361005);