Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 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)(5,6,10,9)(7,12,8,11)(13,14,17,23,18,22,19,24)(15,25,28,36,27,20,29,35)(16,30,33,21,32,38,31,37)(26,40,43,39,41,44,42,34);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15,18,27)(14,20,22,25)(16,26,32,41)(17,28,19,29)(21,34,37,39)(23,35,24,36)(30,44,38,40)(31,42,33,43);(1,3,4,2)(5,6,10,9)(7,12,8,11)(13,16,18,32)(14,21,22,37)(15,26,27,41)(17,31,19,33)(20,34,25,39)(23,30,24,38)(28,42,29,43)(35,44,36,40);(5,8,10,7)(6,11,9,12)(14,22)(15,27)(21,37)(23,24)(26,41)(28,29)(30,38)(42,43);(5,9,10,6)(7,12,8,11)(13,17,18,19)(14,23,22,24)(15,28,27,29)(16,31,32,33)(20,35,25,36)(21,30,37,38)(26,42,41,43)(34,44,39,40)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461743);