Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2280 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14682 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1273 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)(3,15,20,9)(4,21,25,36)(5,26,28,38)(6,13)(7,14)(8,12)(10,23,35,24)(11,27,37,29)(16,41)(17,46)(18,33,19,32)(22,52)(30,43)(31,47)(34,54)(39,44)(40,49)(42,57,62,59)(45,50)(48,56)(51,55)(53,58)(60,61,64,63);(1,3,6,18,8,20,7,19)(2,9,13,32,12,15,14,33)(4,16,23,43,25,45,24,44)(5,17,27,47,28,48,29,49)(10,30,21,50,35,39,36,41)(11,31,26,56,37,40,38,46)(22,42,53,61,54,62,55,63)(34,57,51,64,52,59,58,60);(1,4)(2,10,12,35)(3,16)(5,22)(6,23)(7,24)(8,25)(9,30,15,39)(11,34,37,52)(13,21,14,36)(17,42)(18,43)(19,44)(20,45)(26,51,38,58)(27,53)(28,54)(29,55)(31,57,40,59)(32,50,33,41)(46,60,56,64)(47,61)(48,62)(49,63);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,31)(10,34)(12,37)(13,26)(14,38)(15,40)(16,42)(18,47)(19,49)(20,48)(21,51)(23,53)(24,55)(25,54)(30,57)(32,56)(33,46)(35,52)(36,58)(39,59)(41,60)(43,61)(44,63)(45,62)(50,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299524);