Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360226 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2352 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13872 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1543 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)(4,19,23,10)(5,24)(6,28)(7,12)(8,13)(9,17)(11,25)(15,41,44,35)(16,45)(18,34)(20,31,43,58)(21,50,52,53)(22,37,30,38)(26,39)(27,54)(29,33)(32,47)(36,51,48,56)(40,46)(42,57,63,64)(49,60,62,61)(55,59);(1,3,6,17,8,18,7,29)(2,9,12,14,13,33,28,34)(4,20,22,15,23,43,30,44)(5,16,25,46,27,47,26,55)(10,35,37,58,19,41,38,31)(11,32,24,40,39,45,54,59)(21,49,51,42,52,62,56,63)(36,60,53,64,48,61,50,57);(1,4)(2,10)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31)(11,36)(12,37)(13,19)(14,35)(16,42)(17,43)(18,44)(20,29)(24,53)(25,51)(26,56)(27,52)(28,38)(32,57)(33,58)(34,41)(39,48)(40,60)(45,64)(46,62)(47,63)(49,55)(50,54)(59,61);(1,5,6,25,8,27,7,26)(2,11,12,24,13,39,28,54)(3,16,17,46,18,47,29,55)(4,21,22,51,23,52,30,56)(9,32,14,40,33,45,34,59)(10,36,37,53,19,48,38,50)(15,42,43,62,44,63,20,49)(31,57,35,60,58,64,41,61)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267422);