Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14)(15,23)(16,28)(17,32)(18,21)(19,26)(20,30)(22,25)(24,39)(27,38)(29,37)(31,36)(33,44)(34,42)(35,41)(40,43);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,17,26,18,27,25,42)(14,19,22,23,21,34,32,38)(16,29,31,44,30,24,40,41)(20,35,36,39,28,33,43,37);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,24)(17,31)(18,30)(19,33)(21,28)(22,36)(23,37)(25,40)(26,41)(27,29)(32,43)(34,35)(38,39)(42,44);(1,3,4,2)(7,12)(10,11)(14,21)(16,30)(19,34)(22,32)(23,38)(24,29)(31,40)(41,44);(13,15,25,42,18,27,17,26)(14,19,32,38,21,34,22,23)(16,24,40,44,30,29,31,41)(20,33,43,39,28,35,36,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511546);