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 | |
5815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,7)(3,8,5,4)(9,10,14,12)(11,16,13,15)(17,18,21,23)(19,20,24,22);(2,7)(3,5)(17,19,21,24)(18,22,23,20);(2,7)(4,8)(9,11,14,13)(10,15,12,16)(17,19)(18,20)(21,24)(22,23);(1,3,6,5)(2,4,7,8)(9,11,14,13)(10,16,12,15)(17,20)(18,19)(21,22)(23,24);(1,4,6,8)(2,3,7,5)(9,12,14,10)(11,16,13,15);(1,5,6,3)(2,8,7,4)(9,13,14,11)(10,16,12,15)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72328499 |
| maximal normal subgroup of | #72328499 |
| maximal quotient of | #72328565, #72329272, #72329486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10256321);