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 | |
5690 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)(3,4)(5,6)(7,8)(10,13)(11,12)(19,23)(21,24);(2,6)(3,8)(9,10,14,13)(11,16,12,15)(17,18,20,22)(19,21,23,24);(2,6)(4,7)(9,11,14,12)(10,15,13,16)(17,19)(18,21)(20,23)(22,24);(1,3)(2,7)(4,6)(5,8)(9,12,14,11)(10,16,13,15)(17,18)(19,21)(20,22)(23,24);(1,3)(2,7)(4,6)(5,8)(9,13,14,10)(11,16,12,15)(17,19)(18,21)(20,23)(22,24);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297213, #72300313, #72312153 |
| maximal normal subgroup of | #72297213, #72300313, #72312153 |
| maximal quotient of | #72297258, #72297441, #72297758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8177059);