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 | |
5216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(4,8)(6,7)(10,14)(11,12)(17,18)(19,23)(20,22)(21,24);(2,3)(6,7)(9,10,13,14)(11,15,12,16)(17,18,21,24)(19,22,20,23);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,15,14,16);(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,22,24,23);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=gh,e^2=i,f^2=g,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.
| maximal subgroup of | #72239743 |
| maximal normal subgroup of | #72239743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017175);