Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376201 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14175 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 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)(3,14)(4,20,25,38)(5,26)(6,29)(7,12)(8,13)(9,17)(10,24,35,23)(11,27)(15,31,21,51)(16,45)(18,33)(19,41)(22,53,55,34)(28,36)(30,37)(32,47)(39,44,49,43)(40,46)(42,61,52,60)(48,59)(50,56,57,54)(58,63,64,62);(1,3,6,17,8,19,7,18)(2,9,12,14,13,33,29,41)(4,21,23,44,25,15,24,43)(5,16,27,46,28,47,30,48)(10,31,20,49,35,51,38,39)(11,32,26,40,37,45,36,59)(22,52,54,63,55,42,56,62)(34,58,50,61,53,64,57,60);(1,4)(2,10)(3,15)(5,22)(6,24)(7,23)(8,25)(9,31)(11,34)(12,38)(13,35)(14,39)(16,42)(17,44)(18,43)(19,21)(20,29)(26,57)(27,56)(28,55)(30,54)(32,58)(33,51)(36,50)(37,53)(40,60)(41,49)(45,64)(46,63)(47,52)(48,62)(59,61);(1,5,7,30,8,28,6,27)(2,11,29,36,13,37,12,26)(3,16,18,48,19,47,17,46)(4,22,23,54,25,55,24,56)(9,32,41,59,33,45,14,40)(10,34,20,50,35,53,38,57)(15,42,43,62,21,52,44,63)(31,58,49,61,51,64,39,60)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=ehi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283398);