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 | |
7125 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,20,24,23);(1,3,5,4)(2,6,7,8)(11,12)(14,16)(18,22)(20,23);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,22)(21,24);(2,7)(6,8)(9,12,13,11)(10,14,15,16)(17,20,21,23)(18,19,22,24);(1,4,5,3)(2,8,7,6);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=i,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72241226, #72241230, #72241238, … |
| maximal normal subgroup of | #72241226, #72241230, #72241238, … |
| maximal quotient of | #72263591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8363999);