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 | |
5674 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,19)(20,23,24,22);(2,6)(3,8)(9,11,12,16)(10,13,14,15)(18,19)(22,23);(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,24)(18,22,19,23);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72311959 |
| maximal normal subgroup of | #72311959 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412275);