Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | |
2185 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,20,27,38)(16,29,33,21)(17,34,31,22)(19,24,32,41)(25,43,44,36)(26,40,30,37)(28,39,35,42);(6,9)(10,12)(13,15)(14,20)(16,30)(17,25)(18,27)(19,35)(21,40)(22,36)(23,38)(24,42)(26,33)(28,32)(29,37)(31,44)(34,43)(39,41);(2,4)(13,16)(14,21)(15,25)(17,32)(18,33)(19,31)(20,36)(22,41)(23,29)(24,34)(26,28)(27,44)(30,35)(37,39)(38,43)(40,42);(5,7,8,11)(6,10,9,12)(16,31)(17,33)(19,32)(21,34)(22,29)(24,41)(25,26)(28,35)(30,44)(36,37)(39,42)(40,43);(13,17,18,31)(14,22,23,34)(15,26,27,30)(16,32,33,19)(20,37,38,40)(21,41,29,24)(25,28,44,35)(36,39,43,42)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482369);