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 | |
6254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,14,15)(11,12,16,13);(3,4)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,23);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,11,15,16)(17,19)(18,20)(21,22)(23,24);(3,4)(6,8)(10,15)(12,13)(19,22)(20,24);(1,4,5,3)(2,6,7,8)(9,13,14,12)(10,11,15,16)(17,20,21,24)(18,22,23,19);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232449, #72256120, #72262854, … |
| maximal normal subgroup of | #72232449, #72256120, #72262854, … |
| maximal quotient of | #72232501, #72232548, #72263180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9068069);