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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1802 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)(6,10)(7,8)(13,14,19,25)(15,26,29,21)(16,22,27,35)(17,31,20,23)(18,24,33,40)(28,41,43,36)(30,37,42,39)(32,44,34,38);(5,6)(7,12)(8,11)(9,10)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,30,33,42)(22,23,35,31)(24,39,40,37)(28,34,43,32)(36,38,41,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,20,15,19,27,17,29)(14,22,23,26,25,35,31,21)(18,30,34,28,33,42,32,43)(24,37,38,41,40,39,44,36);(5,8,9,7)(6,11,10,12)(13,17,19,20)(14,23,25,31)(15,27,29,16)(18,32,33,34)(21,35,26,22)(24,38,40,44)(28,42,43,30)(36,39,41,37);(13,18,20,34,19,33,17,32)(14,24,23,38,25,40,31,44)(15,28,27,42,29,43,16,30)(21,36,22,37,26,41,35,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=fhi,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, 6717814);