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 | |
7742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(18,20)(21,23);(2,5)(6,8)(10,13)(14,15)(18,20)(19,24);(1,2)(3,6)(4,5)(7,8)(17,18,22,20)(19,23,24,21);(9,11)(10,14)(12,16)(13,15)(17,19,22,24)(18,23,20,21);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,21,24,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,19)(20,24)(22,23)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267827 |
| maximal subgroup of | #72232197, #72233460, #72233537, … |
| maximal normal subgroup of | #72232197, #72233460, #72233537, … |
| maximal quotient of | #72232268, #72232703, #72263256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015521);