Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,28)(24,35)(25,33)(27,38)(29,39)(30,36)(31,42)(34,41)(37,43)(40,44);(2,3)(5,6)(7,10)(8,12)(9,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,25,30,31)(21,26,35,39)(22,33,36,42)(27,37,40,41)(34,38,43,44);(1,2,4,3)(13,16,15,24,19,29,18,28)(14,21,32,39,23,35,20,26)(17,27,25,37,30,40,31,41)(22,34,42,44,36,43,33,38);(5,7,9,8)(6,10,11,12)(13,17,19,30)(14,22,23,36)(15,25,18,31)(16,27,29,40)(20,33,32,42)(21,34,35,43)(24,37,28,41)(26,38,39,44);(1,3,4,2)(5,8,9,7)(6,10,11,12)(17,30)(22,36)(25,31)(27,40)(33,42)(34,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6724219);