Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,27,36)(16,30,33,37)(17,23,18,22)(20,29,35,28)(21,31,38,32)(26,40,41,34)(39,42,44,43);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,26)(17,29)(18,28)(19,27)(21,34)(22,36)(23,25)(24,35)(30,44)(31,43)(32,42)(33,41)(37,39)(38,40);(13,16,19,33)(14,21,24,38)(15,26,27,41)(17,32,18,31)(20,34,35,40)(22,30,23,37)(25,39,36,44)(28,43,29,42);(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536436);