Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467073 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2640 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16558 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,29,8,12,7,13)(3,14,19,42,17,9,18,33)(4,20,23,36,24,10,30,37)(5,25,28,39,26,56,27,11)(15,40,49,50,44,31,21,34)(16,45,46,41,48,55,47,32)(22,52,57,38,53,61,54,35)(43,60,51,58,63,62,64,59);(1,3,6,18,8,17,7,19)(2,9,29,42,12,14,13,33)(4,21,23,44,24,49,30,15)(5,16,27,46,26,48,28,47)(10,34,37,31,20,50,36,40)(11,32,56,45,39,41,25,55)(22,51,54,63,53,64,57,43)(35,60,61,58,38,62,52,59);(1,4)(2,10)(3,15)(5,22)(6,30)(7,23)(8,24)(9,31)(11,38)(12,20)(13,37)(14,40)(16,43)(17,44)(18,49)(19,21)(25,52)(26,53)(27,57)(28,54)(29,36)(32,59)(33,50)(34,42)(35,39)(41,58)(45,62)(46,64)(47,51)(48,63)(55,60)(56,61);(1,5)(2,11,12,39)(3,16)(4,22)(6,27)(7,28)(8,26)(9,32,14,41)(10,35,20,38)(13,25,29,56)(15,43)(17,48)(18,46)(19,47)(21,51)(23,54)(24,53)(30,57)(31,58,40,59)(33,55,42,45)(34,60,50,62)(36,52,37,61)(44,63)(49,64)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374269);