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 | #42788 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 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)(6,9)(7,11)(13,14,19,25)(15,26,29,21)(16,22,27,36)(17,31,20,23)(18,33,34,24)(28,37,43,41)(30,40,42,38)(32,39,35,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,30,34,42)(22,23,36,31)(24,40,33,38)(28,35,43,32)(37,39,41,44);(2,4)(13,16,20,15,19,27,17,29)(14,22,23,26,25,36,31,21)(18,30,35,28,34,42,32,43)(24,38,39,41,33,40,44,37);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,31)(15,27,29,16)(18,32,34,35)(21,36,26,22)(24,39,33,44)(28,42,43,30)(37,40,41,38);(6,9)(10,12)(13,18,17,32,19,34,20,35)(14,24,31,44,25,33,23,39)(15,28,16,30,29,43,27,42)(21,37,36,40,26,41,22,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,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, 6715146);