Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13957 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
987 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)(3,14,18,33)(4,19)(5,24,27,54)(6,12)(7,28)(8,13)(9,17,34,29)(10,22)(11,25,38,26)(15,41,20,31)(16,45)(21,35,52,59)(23,36)(30,37)(32,47)(39,43,48,44)(40,55)(42,62)(46,58)(49,51,53,56)(50,60)(57,63)(61,64);(1,3,7,29,8,18,6,17)(2,9,12,14,13,34,28,33)(4,20,30,43,23,15,22,44)(5,16,26,55,27,47,25,46)(10,31,36,39,37,41,19,48)(11,32,24,40,38,45,54,58)(21,50,56,63,52,42,51,64)(35,57,53,60,59,61,49,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31)(11,35)(12,36)(13,37)(14,39)(16,42)(17,43)(18,20)(19,28)(24,53)(25,51)(26,56)(27,52)(29,44)(32,57)(33,48)(34,41)(38,59)(40,60)(45,61)(46,63)(47,50)(49,54)(55,64)(58,62);(1,5,8,27)(2,11)(3,16,18,47)(4,21,23,52)(6,25,7,26)(9,32)(10,35)(12,24)(13,38)(14,40)(15,42,20,50)(17,46,29,55)(19,49)(22,51,30,56)(28,54)(31,57)(33,58)(34,45)(36,53)(37,59)(39,60)(41,61)(43,63,44,64)(48,62)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271213);