Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467842 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16592 |
| 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 | |
794 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)(3,15)(4,21,26,42)(5,11,30,40)(6,13)(7,12)(8,14)(9,18)(10,25,37,24)(16,44,47,31)(17,48,52,43)(19,34)(20,33)(22,54,46,35)(23,53)(27,57)(28,41,29,39)(32,51,61,50)(36,55)(38,56)(45,59)(49,62)(58,63)(60,64);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,25,47,26,46,24,16)(5,17,29,51,30,52,28,50)(10,35,42,31,37,54,21,44)(11,32,41,43,40,61,39,48)(23,49,27,58,56,64,55,45)(36,62,38,59,57,60,53,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,37,13,42)(3,16,18,46,20,47,19,22)(5,27,28,56,30,55,29,23)(9,31,15,35,34,44,33,54)(11,38,39,57,40,53,41,36)(17,49,50,58,52,64,51,45)(32,60,48,63,61,62,43,59);(1,5,8,30)(2,11)(3,17,20,52)(4,23,26,56)(6,29,7,28)(9,32)(10,36)(12,41)(13,39)(14,40)(15,43)(16,45,47,58)(18,51,19,50)(21,53)(22,49,46,64)(24,55,25,27)(31,59)(33,48)(34,61)(35,62)(37,57)(38,42)(44,63)(54,60)Polycyclic presentation:
2^9|a^2=fg,b^2=efg,c^2=e,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375039);