Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14)(15,23)(16,21)(17,19)(18,22)(20,26)(24,33)(25,37)(27,31)(28,36)(29,32)(30,35)(34,40)(38,43)(39,44)(41,42);(6,9)(7,12)(14,19)(15,24)(16,27)(18,29)(20,33)(23,35)(25,39)(26,30)(34,44)(36,42)(37,43)(38,40);(2,4)(13,15,17,26)(14,20,19,23)(16,25,27,40)(18,30,29,24)(21,34,31,37)(22,33,32,35)(28,39,41,38)(36,43,42,44);(2,4)(5,7,8,12)(6,10,9,11)(13,16,17,27)(14,21,19,31)(15,25,26,40)(18,28,29,41)(20,34,23,37)(22,36,32,42)(24,38,30,39)(33,43,35,44);(16,27)(21,31)(25,40)(28,41)(34,37)(36,42)(38,39)(43,44)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409602);