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 |
| prime radical | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,27,36)(16,21,31,40)(17,22,18,23)(25,34,29,38)(26,35,32,41)(28,37,43,44)(30,39,33,42);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,28)(17,26)(18,32)(19,27)(21,37)(22,35)(23,41)(24,36)(25,30)(29,33)(31,43)(34,39)(38,42)(40,44);(6,10)(7,11)(13,15)(14,20)(16,29)(17,32)(18,26)(19,27)(21,38)(22,41)(23,35)(24,36)(25,31)(28,33)(30,43)(34,40)(37,42)(39,44);(6,10)(9,12)(13,16,18,30,19,31,17,33)(14,21,23,39,24,40,22,42)(15,25,26,28,27,29,32,43)(20,34,35,37,36,38,41,44);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,26,27,32)(16,30,31,33)(20,35,36,41)(21,39,40,42)(25,43,29,28)(34,44,38,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7518754);