Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13941 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,20,32,19,33,18,9)(4,10,23,37,24,38,29,36)(5,25,28,39,27,40,26,11)(16,42,21,30,44,34,49,57)(17,45,47,58,48,31,46,41)(22,51,52,35,53,54,55,60)(43,62,50,61,63,56,64,59);(1,3,6,18,8,19,7,20)(2,9,12,15,14,32,13,33)(4,21,23,44,24,49,29,16)(5,17,26,46,27,48,28,47)(10,34,36,30,38,42,37,57)(11,31,25,41,39,45,40,58)(22,50,52,63,53,64,55,43)(35,59,51,56,60,61,54,62);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,49)(19,44)(20,21)(25,54)(26,55)(27,53)(28,52)(31,56)(32,57)(33,42)(39,60)(40,51)(41,59)(45,62)(46,64)(47,50)(48,63)(58,61);(1,5,8,27)(2,11)(3,17,19,48)(4,22,24,53)(6,26,7,28)(9,31)(10,35)(12,25)(13,40)(14,39)(15,41)(16,43,44,63)(18,46,20,47)(21,50,49,64)(23,52,29,55)(30,56)(32,45)(33,58)(34,59)(36,51)(37,54)(38,60)(42,61)(57,62)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270703);