Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15324 |
| 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 | |
1044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,15,18,33,20,9,19,34)(4,21,24,10,25,37,26,36)(5,27,28,11,29,40,30,39)(16,42,44,41,22,51,45,31)(17,46,47,38,48,57,49,32)(23,53,54,50,55,35,56,60)(43,58,62,61,52,59,63,64);(1,3,8,20)(2,9,12,15)(4,22,25,16)(5,17,29,48)(6,19,7,18)(10,31,36,41)(11,38,39,32)(13,34,14,33)(21,42,37,51)(23,52,55,43)(24,44,26,45)(27,57,40,46)(28,49,30,47)(35,59,53,58)(50,64,60,61)(54,62,56,63);(1,4,6,24,8,25,7,26)(2,10,13,21,12,36,14,37)(3,16,18,44,20,22,19,45)(5,23,28,54,29,55,30,56)(9,31,33,51,15,41,34,42)(11,35,27,50,39,53,40,60)(17,43,47,62,48,52,49,63)(32,58,57,64,38,59,46,61);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,35)(12,39)(13,27)(14,40)(15,38)(16,43)(18,47)(19,49)(20,48)(21,50)(22,52)(24,54)(25,55)(26,56)(31,58)(33,57)(34,46)(36,53)(37,60)(41,59)(42,61)(44,62)(45,63)(51,64)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 331930);