Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16309 |
| 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 | |
883 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)(3,15,18,9)(4,20,25,36)(5,26,29,38)(6,13,7,12)(10,24,37,23)(11,28,39,27)(16,34,44,59)(17,46,48,57)(19,33,30,40)(21,31,45,41)(22,53,55,35)(32,50,42,49)(43,60,47,58)(51,56,62,54)(52,61,63,64);(1,3,6,30,8,18,7,19)(2,9,12,40,14,15,13,33)(4,21,23,44,25,45,24,16)(5,17,27,50,29,48,28,49)(10,34,20,41,37,59,36,31)(11,32,26,57,39,42,38,46)(22,52,54,47,55,63,56,43)(35,61,51,60,53,64,62,58);(1,4,6,23,8,25,7,24)(2,10,12,20,14,37,13,36)(3,16,30,21,18,44,19,45)(5,22,27,54,29,55,28,56)(9,31,40,34,15,41,33,59)(11,35,26,51,39,53,38,62)(17,47,50,63,48,43,49,52)(32,60,57,64,42,58,46,61);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,29)(9,32)(10,35)(12,26)(13,38)(14,39)(15,42)(16,43)(18,48)(19,49)(20,51)(21,52)(23,54)(24,56)(25,55)(30,50)(31,58)(33,46)(34,61)(36,62)(37,53)(40,57)(41,60)(44,47)(45,63)(59,64)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=ef,f^2=i,g^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,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365269);