Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,21)(17,20)(18,22)(23,39)(24,33)(25,34)(26,32)(27,37)(28,35)(29,38)(30,36)(31,40)(41,44)(42,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,20,34)(16,27,28,42)(18,23,30,26)(21,37,35,43)(22,39,36,32)(24,31,41,29)(33,40,44,38);(1,2,3,4)(6,9)(10,11)(14,20)(15,23)(18,30)(19,32)(21,35)(24,27)(25,26)(29,31)(33,43)(34,39)(37,44)(41,42);(5,7,8,12)(6,10,9,11)(16,28)(21,35)(24,41)(27,42)(29,31)(33,44)(37,43)(38,40);(13,16,17,28)(14,21,20,35)(15,24,25,41)(18,31,30,29)(19,33,34,44)(22,40,36,38)(23,27,26,42)(32,43,39,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592205);