Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,18,32)(16,26,29,40)(17,22,31,37)(21,24,35,28)(25,33,30,36)(27,39,42,43)(34,38,44,41);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,30,31,25)(21,26,35,40)(22,36,37,33)(27,41,42,38)(34,43,44,39);(2,4)(13,16,15,24,19,29,18,28)(14,21,32,40,23,35,20,26)(17,27,25,38,31,42,30,41)(22,34,36,43,37,44,33,39);(6,10)(9,11)(13,17,19,31)(14,22,23,37)(15,25,18,30)(16,27,29,42)(20,33,32,36)(21,34,35,44)(24,38,28,41)(26,39,40,43);(2,4)(7,12)(9,11)(17,31)(22,37)(25,30)(27,42)(33,36)(34,44)(38,41)(39,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589605);