Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14)(15,25)(16,30)(17,23)(18,24)(19,34)(20,28)(21,31)(22,32)(26,36)(27,41)(29,37)(33,38)(35,44)(39,42)(40,43);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,24,25)(16,26,32,43)(17,27,19,29)(21,35,38,39)(22,36,30,40)(23,37,34,41)(31,42,33,44);(1,2,4,3)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,35)(22,23)(24,38)(25,39)(27,42)(28,43)(29,44)(30,34)(36,37)(40,41);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,33)(18,32)(19,31)(20,36)(21,23)(24,30)(25,40)(27,44)(28,43)(29,42)(34,38)(35,37)(39,41);(1,3,4,2)(6,9)(10,12)(13,17,18,19)(14,23,24,34)(15,27,28,29)(16,31,32,33)(20,37,25,41)(21,22,38,30)(26,42,43,44)(35,36,39,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715502);