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 | |
6221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,13)(11,12)(18,20)(22,24);(1,2)(3,8)(4,6)(5,7)(9,10,14,13)(11,16,12,15)(19,23)(22,24);(1,3,5,6)(2,4,7,8)(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,22,23,24);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4)(5,7)(6,8)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,24,23,22);(1,4)(2,6)(3,7)(5,8)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232576, #72249674, #72256065, … |
| maximal normal subgroup of | #72232576, #72249674, #72256065, … |
| maximal quotient of | #72232600, #72263207, #72326425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8572148);