Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 |
| prime radical | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,11,10)(7,8,9,12)(13,14,19,23)(15,24,27,41)(16,21,31,39)(17,22,18,33)(20,26,35,28)(25,40,29,44)(30,37,32,38)(34,43,36,42);(5,7,11,9)(6,12,10,8)(13,15,18,28,19,27,17,26)(14,20,22,24,23,35,33,41)(16,29,30,43,31,25,32,42)(21,36,38,44,39,34,37,40);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,38)(23,39)(24,40)(26,43)(27,29)(28,42)(33,37)(35,36)(41,44);(5,9,11,7)(6,12,10,8)(13,17,19,18)(14,22,23,33)(15,26,27,28)(16,30,31,32)(20,24,35,41)(21,37,39,38)(25,42,29,43)(34,44,36,40);(5,10,11,6)(7,8,9,12)(15,27)(20,35)(24,41)(25,29)(26,28)(34,36)(40,44)(42,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477795);