Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 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)(16,31)(17,33)(18,35)(19,25)(20,23)(21,30)(22,32)(24,36)(27,39)(28,40)(29,41)(34,43)(37,44)(38,42);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,18)(17,28)(19,30)(20,38)(22,35)(23,40)(24,31)(25,26)(27,29)(32,36)(33,42)(34,37)(39,43)(41,44);(2,4)(13,16,20,29,19,32,17,34)(14,22,23,43,25,31,33,41)(15,27,28,18,30,37,38,36)(21,39,42,35,26,44,40,24);(6,10)(9,11)(13,17,19,20)(14,23,25,33)(15,28,30,38)(16,29,32,34)(18,27,36,37)(21,40,26,42)(22,41,31,43)(24,39,35,44);(7,12)(9,11)(13,18,17,27,19,36,20,37)(14,24,33,44,25,35,23,39)(15,29,38,16,30,34,28,32)(21,41,40,31,26,43,42,22)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,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6713870);