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 | |
6716 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,12,14,16)(17,18)(19,22)(20,23)(21,24);(2,5)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,21,20)(18,22,24,23);(1,2)(3,7)(4,5)(6,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,21)(22,24);(17,18)(19,22)(20,23)(21,24);(1,3,4,8)(2,6,5,7)(9,11)(10,12)(13,14)(15,16)(17,20,21,19)(18,23,24,22);(1,2)(3,7)(4,5)(6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,e^2=gh,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi125 conjugacy classes — view the full character table.
| maximal subgroup of | #72233516, #72233523, #72233607, … |
| maximal normal subgroup of | #72233516, #72233523, #72233607, … |
| maximal quotient of | #72261144, #72262157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7606946);