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 | |
6058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,5)(3,7,8,4)(9,10,12,14)(11,13,16,15);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,23);(9,10)(11,13)(12,14)(15,16)(17,19,21,22)(18,20,23,24);(2,5)(3,8)(9,10)(11,15)(12,14)(13,16)(17,20,21,24)(18,19,23,22);(1,4)(2,3)(5,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,20)(21,23)(22,24);(1,5,6,2)(3,7,8,4)(9,11,12,16)(10,13,14,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72258018, #72283347, #72285863, … |
| maximal normal subgroup of | #72258018, #72283347, #72285863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604515);