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 | |
6052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,5,7)(10,13)(11,15)(17,18)(19,21)(20,23)(22,24);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(18,19)(22,23);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,19)(20,22,24,23);(2,4)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72238563, #72238566, #72238573 |
| maximal normal subgroup of | #72238563, #72238566, #72238573 |
| maximal quotient of | #72262096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8079858);