Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15385 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,26,7,12,29,11)(3,13,16,8,17,42,49,32)(4,18,21,34,22,9,27,38)(5,10,24,39,25,37,28,36)(14,43,45,30,19,40,47,50)(15,41,23,57,46,35,48,31)(20,51,54,60,55,33,56,61)(44,62,63,64,52,59,53,58);(1,3,6,16,7,17,29,49)(2,8,11,13,12,32,26,42)(4,19,21,47,22,14,27,45)(5,23,24,46,25,48,28,15)(9,30,34,43,18,50,38,40)(10,35,36,57,37,41,39,31)(20,53,54,44,55,63,56,52)(33,59,60,64,51,62,61,58);(1,4,7,22)(2,9,12,18)(3,14,17,19)(5,20,25,55)(6,27,29,21)(8,30,32,50)(10,33,37,51)(11,38,26,34)(13,40,42,43)(15,44,46,52)(16,47,49,45)(23,53,48,63)(24,56,28,54)(31,58,57,64)(35,62,41,59)(36,61,39,60);(1,5)(2,10,12,37)(3,15)(4,20)(6,28)(7,25)(8,31,32,57)(9,33,18,51)(11,39,26,36)(13,41,42,35)(14,44)(16,48)(17,46)(19,52)(21,56)(22,55)(23,49)(24,29)(27,54)(30,58,50,64)(34,61,38,60)(40,59,43,62)(45,53)(47,63)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,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, 335761);