Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427831 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15375 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,18,33)(4,19,22,37)(5,25,27,11)(6,13,7,12)(10,24,36,23)(15,34,42,60)(16,43,46,31)(17,39,28,32)(20,30,47,48)(21,51,53,49)(26,38,29,40)(35,54,56,55)(41,62,63,61)(44,58,45,59)(50,64,52,57);(1,3)(2,9)(4,20)(5,16)(6,17)(7,28)(8,18)(10,34)(11,31)(12,32)(13,39)(14,33)(15,24)(19,48)(21,52)(22,47)(23,42)(25,43)(26,44)(27,46)(29,45)(30,37)(35,62)(36,60)(38,58)(40,59)(41,54)(49,57)(50,53)(51,64)(55,63)(56,61);(1,4)(2,10)(3,15)(5,21)(6,24)(7,23)(8,22)(9,30)(11,35)(12,37)(13,19)(14,36)(16,41)(17,47)(18,42)(20,28)(25,56)(26,55)(27,53)(29,54)(31,57)(32,60)(33,48)(34,39)(38,51)(40,49)(43,64)(44,52)(45,50)(46,63)(58,62)(59,61);(1,5,6,26,8,27,7,29)(2,11,12,38,14,25,13,40)(3,16,17,44,18,46,28,45)(4,21,24,55,22,53,23,54)(9,31,32,58,33,43,39,59)(10,35,37,51,36,56,19,49)(15,41,47,52,42,63,20,50)(30,57,60,62,48,64,34,61)Polycyclic presentation:
2^9|a^2=i,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 335028);