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 | |
5646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,16)(12,14)(18,20)(22,23);(2,5)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18,21,20)(19,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,22,20,23);(9,10)(11,12)(13,15)(14,16)(17,20,21,18)(19,23,24,22);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72297120 |
| maximal normal subgroup of | #72297120 |
| maximal quotient of | #72297156, #72297247, #72297817 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9687734);