Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(7,11)(8,10)(9,12)(13,14)(15,26)(16,21)(17,25)(18,23)(19,24)(20,29)(22,31)(27,41)(28,35)(30,38)(32,40)(33,39)(34,43)(36,42)(37,44);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,30)(17,28)(18,31)(19,29)(21,36)(22,35)(23,25)(24,26)(27,32)(33,42)(34,40)(37,41)(38,39)(43,44);(1,2,3,4)(13,16,31,44,19,33,17,32)(14,21,22,37,24,39,25,40)(15,27,28,42,29,43,18,30)(20,34,23,38,26,41,35,36);(6,10)(9,11)(13,17,19,31)(14,22,24,25)(15,28,29,18)(16,32,33,44)(20,35,26,23)(21,37,39,40)(27,42,43,30)(34,36,41,38);(7,12)(9,11)(13,18,17,29,19,28,31,15)(14,23,25,20,24,35,22,26)(16,30,32,43,33,42,44,27)(21,38,40,34,39,36,37,41)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721591);