Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2310 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14346 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
821 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)(3,9)(4,19,22,50)(5,11)(6,12)(7,29)(8,13)(10,23,36,24)(14,34,41,62)(15,31)(16,32)(17,47)(18,33)(20,30,42,48)(21,49,25,54)(26,38)(27,56)(28,39)(35,52,37,53)(40,61,43,63)(44,59)(45,64)(46,60)(51,57,55,58);(1,3)(2,9)(4,20,22,42)(5,15)(6,16)(7,17)(8,18)(10,34,36,62)(11,31)(12,32)(13,33)(14,24,41,23)(19,48,50,30)(21,51,25,55)(26,44)(27,45)(28,46)(29,47)(35,61,37,63)(38,59)(39,60)(40,53,43,52)(49,58,54,57)(56,64);(1,4)(2,10,13,36)(3,14,18,41)(5,25,28,21)(6,23)(7,24)(8,22)(9,30)(11,37)(12,19,29,50)(15,43)(16,42,17,20)(26,53,27,52)(31,58,60,57)(32,34)(33,48)(35,39)(38,54)(40,46)(44,51)(45,55)(47,62)(49,56)(59,63,64,61);(1,5,6,26,8,28,7,27)(2,11,12,38,13,39,29,56)(3,15,16,44,18,46,17,45)(4,21,23,52,22,25,24,53)(9,31,32,59,33,60,47,64)(10,35,19,49,36,37,50,54)(14,40,42,55,41,43,20,51)(30,57,34,61,48,58,62,63)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=h,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290097);