Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,26)(16,22)(17,32)(18,24)(19,25)(20,23)(21,30)(27,44)(28,38)(29,43)(31,42)(33,39)(34,41)(35,37)(36,40);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,18)(17,28)(19,30)(20,36)(22,41)(23,38)(24,42)(25,26)(27,29)(31,34)(32,40)(33,35)(37,43)(39,44);(2,4)(13,16,20,29,19,31,17,33)(14,22,23,43,25,42,32,39)(15,27,28,18,30,35,36,34)(21,37,40,41,26,44,38,24);(6,10)(9,11)(13,17,19,20)(14,23,25,32)(15,28,30,36)(16,29,31,33)(18,27,34,35)(21,38,26,40)(22,39,42,43)(24,37,41,44);(7,12)(9,11)(13,18,17,27,19,34,20,35)(14,24,32,44,25,41,23,37)(15,29,36,16,30,33,28,31)(21,39,38,42,26,43,40,22)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6713822);