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 | |
6532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(17,18)(19,21)(20,24)(22,23);(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(18,23)(19,24);(1,2,5,4)(3,6,7,8)(10,15)(11,12)(17,19,22,24)(18,20,23,21);(9,11,14,12)(10,16,15,13)(17,20,22,21)(18,19,23,24);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,13,15,16)(17,18)(19,20)(21,24)(22,23);(1,4,5,2)(3,8,7,6)(9,13)(10,11)(12,15)(14,16)(17,21,22,20)(18,24,23,19)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256065, #72258018, #72262887, … |
| maximal normal subgroup of | #72256065, #72258018, #72262887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706309);