Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,24,26,20)(16,28,29,37)(17,23,19,33)(21,32,38,30)(25,40,42,44)(27,36,31,35)(34,43,39,41);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,20,22,24)(16,25,29,42)(17,31,19,27)(21,34,38,39)(23,36,33,35)(28,44,37,40)(30,43,32,41);(2,4)(6,9)(10,11)(13,16,17,32,18,29,19,30)(14,21,33,37,22,38,23,28)(15,25,31,41,26,42,27,43)(20,34,35,40,24,39,36,44);(5,7,8,12)(6,10,9,11)(14,22)(15,26)(16,29)(23,33)(27,31)(30,32)(34,39)(40,44);(2,4)(13,17,18,19)(14,23,22,33)(15,27,26,31)(16,30,29,32)(20,35,24,36)(21,37,38,28)(25,41,42,43)(34,44,39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592143);