Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2428 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13,7,14,6,12)(3,15,19,9,20,42,18,33)(4,21,29,36,24,37,25,10)(5,11,30,40,28,41,27,39)(16,34,47,43,22,31,46,50)(17,44,49,32,26,55,48,38)(23,51,56,59,53,60,54,35)(45,58,63,61,52,57,62,64);(1,3)(2,9)(4,22)(5,26)(6,19)(7,20)(8,18)(10,34)(11,38)(12,42)(13,15)(14,33)(16,24)(17,28)(21,50)(23,52)(25,46)(27,48)(29,47)(30,49)(31,36)(32,41)(35,58)(37,43)(39,44)(40,55)(45,53)(51,64)(54,62)(56,63)(57,59)(60,61);(1,4,7,24)(2,10)(3,16)(5,23,28,53)(6,25,8,29)(9,31,33,34)(11,35)(12,37)(13,21)(14,36)(15,43,42,50)(17,45)(18,46)(19,47)(20,22)(26,52)(27,54,30,56)(32,57,38,58)(39,60)(40,51)(41,59)(44,61,55,64)(48,62)(49,63);(1,5,6,27,7,28,8,30)(2,11,12,39,14,41,13,40)(3,17,18,48,20,26,19,49)(4,23,25,54,24,53,29,56)(9,32,15,44,33,38,42,55)(10,35,37,60,36,59,21,51)(16,45,46,62,22,52,47,63)(31,57,43,61,34,58,50,64);(4,24)(10,36)(16,22)(21,37)(23,53)(25,29)(31,34)(35,59)(43,50)(45,52)(46,47)(51,60)(54,56)(57,58)(61,64)(62,63)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351593);