Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #375190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14152 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,14,17,32,18,9,28,33)(4,10,23,36,21,37,22,51)(5,24,26,38,25,11,29,39)(15,40,19,34,43,30,48,55)(16,31,46,56,44,41,45,57)(20,35,50,60,49,53,52,59)(42,61,63,64,62,54,47,58);(1,3,6,17,8,18,7,28)(2,9,27,33,12,14,13,32)(4,19,23,15,21,48,22,43)(5,16,26,45,25,44,29,46)(10,34,36,40,37,55,51,30)(11,31,39,57,24,41,38,56)(20,47,52,42,49,63,50,62)(35,58,59,61,53,64,60,54);(1,4,6,22,8,21,7,23)(2,10,27,51,12,37,13,36)(3,15,28,48,18,43,17,19)(5,20,26,50,25,49,29,52)(9,30,32,34,14,40,33,55)(11,35,39,60,24,53,38,59)(16,42,46,63,44,62,45,47)(31,54,56,58,41,61,57,64);(1,5)(2,11)(3,16)(4,20)(6,26)(7,29)(8,25)(9,31)(10,35)(12,24)(13,38)(14,41)(15,42)(17,45)(18,44)(19,47)(21,49)(22,50)(23,52)(27,39)(28,46)(30,54)(32,56)(33,57)(34,58)(36,59)(37,53)(40,61)(43,62)(48,63)(51,60)(55,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fhi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282389);