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 | |
6955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,15)(12,16)(17,18,21,22)(19,20,24,23);(1,2)(3,4)(5,7)(6,8)(9,10,14,13)(11,16,15,12)(17,18)(19,23)(20,24)(21,22);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22);(3,8)(4,6)(10,13)(11,15)(18,22)(20,23);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24);(1,4,5,6)(2,3,7,8)(9,13,14,10)(11,16,15,12)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|a^2=g,b^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72263262, #72263768 |
| maximal normal subgroup of | #72263262, #72263768 |
| maximal quotient of | #72263273, #72263781 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8628389);