Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15316 |
| 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 | |
1255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,15,18,35,19,34,30,9)(4,20,23,10,24,37,25,38)(5,26,28,11,29,40,31,41)(16,32,44,42,21,49,51,60)(17,45,27,57,47,33,46,39)(22,52,54,48,55,36,56,62)(43,61,58,63,50,64,53,59);(1,3)(2,9)(4,21)(5,27)(6,30)(7,18)(8,19)(10,32)(11,39)(12,34)(13,15)(14,35)(16,24)(17,28)(20,42)(22,53)(23,44)(25,51)(26,45)(29,46)(31,47)(33,40)(36,61)(37,60)(38,49)(41,57)(43,56)(48,63)(50,54)(52,64)(55,58)(59,62);(1,4,6,23,8,24,7,25)(2,10,13,20,14,38,12,37)(3,16,18,44,19,21,30,51)(5,22,28,54,29,55,31,56)(9,32,34,60,35,49,15,42)(11,36,26,48,41,52,40,62)(17,43,27,58,47,50,46,53)(33,59,57,64,45,63,39,61);(1,5,6,28,8,29,7,31)(2,11,12,40,14,41,13,26)(3,17,18,27,19,47,30,46)(4,22,23,54,24,55,25,56)(9,33,15,39,35,45,34,57)(10,36,37,62,38,52,20,48)(16,43,44,58,21,50,51,53)(32,59,42,61,49,63,60,64)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331050);