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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14150 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,32,18,41,17,9)(4,10,23,35,24,37,22,36)(5,25,27,38,28,54,26,11)(15,33,20,30,44,39,43,56)(16,45,48,61,47,31,46,40)(21,50,52,59,53,58,51,34)(42,55,49,60,63,64,62,57);(1,3,6,17,8,18,7,19)(2,9,12,14,13,32,29,41)(4,20,23,44,24,43,22,15)(5,16,26,46,28,47,27,48)(10,33,36,56,37,39,35,30)(11,31,25,40,38,45,54,61)(21,49,52,63,53,62,51,42)(34,57,58,64,59,60,50,55);(1,4,7,23,8,24,6,22)(2,10,12,35,13,37,29,36)(3,15,19,20,18,44,17,43)(5,21,27,52,28,53,26,51)(9,30,14,39,32,56,41,33)(11,34,25,50,38,59,54,58)(16,42,48,49,47,63,46,62)(31,55,40,60,45,64,61,57);(1,5,6,26,8,28,7,27)(2,11,12,25,13,38,29,54)(3,16,17,46,18,47,19,48)(4,21,22,51,24,53,23,52)(9,31,14,40,32,45,41,61)(10,34,35,50,37,59,36,58)(15,42,43,62,44,63,20,49)(30,55,39,60,56,64,33,57)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282266);