Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14886 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1317 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)(3,13)(4,9)(5,21)(6,24)(7,12)(8,16)(10,23)(11,25)(14,37)(15,41)(17,36)(18,30)(19,47)(20,32)(22,33)(26,53)(27,55)(28,44)(29,39)(31,48)(34,40)(35,43)(38,59)(42,49)(45,62)(46,50)(51,52)(54,64)(56,60)(57,63)(58,61);(1,3,6,16,7,17,25,44)(2,8,11,13,12,28,24,36)(4,14,19,39,20,40,48,53)(5,15,22,42,23,43,52,55)(9,29,31,37,32,26,47,34)(10,27,33,35,21,49,51,41)(18,38,45,60,46,61,63,64)(30,56,57,59,50,54,62,58);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,26)(10,30)(11,31)(12,32)(13,34)(15,38)(16,39)(17,40)(21,50)(22,45)(23,46)(24,47)(25,48)(27,54)(28,29)(33,57)(35,58)(36,37)(41,59)(42,60)(43,61)(44,53)(49,56)(51,62)(52,63)(55,64);(1,5)(2,10,12,21)(3,15)(4,18)(6,22)(7,23)(8,27,28,49)(9,30,32,50)(11,33,24,51)(13,35,36,41)(14,38)(16,42)(17,43)(19,45)(20,46)(25,52)(26,54,29,56)(31,57,47,62)(34,58,37,59)(39,60)(40,61)(44,55)(48,63)(53,64)Polycyclic presentation:
2^9|b^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311303);