Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,4,8,6)(10,13)(11,12)(17,18,21,23)(19,20,24,22);(1,3)(2,4)(5,8)(6,7)(9,10,14,13)(11,16,12,15)(17,19,21,24)(18,22,23,20);(1,3,5,8)(2,6,7,4)(9,11,14,12)(10,15,13,16)(17,19)(18,20)(21,24)(22,23);(9,12,14,11)(10,16,13,15)(17,20)(18,19)(21,22)(23,24);(1,4)(2,3)(5,6)(7,8)(9,13,14,10)(11,16,12,15);(1,3)(2,4)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=ghi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8100986);