Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,28,37)(3,13,16,38,17,8,26,31)(4,18,21,39,22,9,27,34)(5,23,25,52,24,10,35,36)(14,29,19,32,43,40,46,47)(15,30,45,56,44,41,53,59)(20,48,51,60,50,33,54,58)(42,55,49,57,62,61,63,64);(1,3,7,17)(2,8,11,13)(4,19,22,46)(5,15,24,44)(6,26,28,16)(9,32,18,47)(10,30,23,41)(12,38,37,31)(14,27,43,21)(20,49,50,63)(25,53,35,45)(29,39,40,34)(33,57,48,64)(36,59,52,56)(42,54,62,51)(55,60,61,58);(1,4)(2,9)(3,14)(5,20)(6,27)(7,22)(8,29)(10,33)(11,18)(12,39)(13,40)(15,42)(16,46)(17,43)(19,26)(21,28)(23,48)(24,50)(25,54)(30,55)(31,47)(32,38)(34,37)(35,51)(36,60)(41,61)(44,62)(45,63)(49,53)(52,58)(56,64)(57,59);(1,5)(2,10,11,23)(3,15)(4,20)(6,25)(7,24)(8,30,13,41)(9,33,18,48)(12,36,37,52)(14,42)(16,45)(17,44)(19,49)(21,51)(22,50)(26,53)(27,54)(28,35)(29,55,40,61)(31,56,38,59)(32,57,47,64)(34,58,39,60)(43,62)(46,63)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308997);