Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,28)(24,39)(25,34)(26,36)(29,37)(30,35)(31,38)(33,41)(40,43)(42,44);(1,2)(3,4)(6,9)(7,11)(13,15,19,26)(14,20,23,36)(16,24,29,42)(17,30,18,25)(21,33,38,43)(22,35,32,34)(27,39,37,44)(28,41,31,40);(2,4)(13,16,18,28,19,29,17,31)(14,21,22,27,23,38,32,37)(15,24,25,41,26,42,30,40)(20,33,35,39,36,43,34,44);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,25,26,30)(16,28,29,31)(20,34,36,35)(21,37,38,27)(24,40,42,41)(33,39,43,44);(2,4)(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592466);