Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,26)(16,22)(17,23)(18,21)(19,35)(20,25)(24,28)(27,39)(29,38)(30,44)(31,41)(32,37)(33,40)(34,36)(42,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,32,20,28,19,18)(14,21,35,24,25,37,23,26)(16,27,33,30,31,43,29,34)(22,36,38,42,41,44,40,39);(6,10)(7,11)(13,16,20,31)(14,22,25,41)(15,27,28,43)(17,33,19,29)(18,34,32,30)(21,36,37,44)(23,40,35,38)(24,42,26,39);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,35)(15,18,28,32)(16,29,31,33)(21,24,37,26)(22,38,41,40)(27,30,43,34)(36,39,44,42);(13,18,19,28,20,32,17,15)(14,24,23,21,25,26,35,37)(16,30,29,27,31,34,33,43)(22,39,40,44,41,42,38,36)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7519007);