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 | |
6288 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,12,15,16)(18,20)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11,14,15)(10,12,13,16)(19,23)(22,24);(9,12)(10,11)(13,15)(14,16)(17,18,21,20)(19,22,23,24);(1,2,4,6)(3,5,8,7)(9,13,14,10)(11,12,15,16)(17,19,21,23)(18,22,20,24);(1,3)(2,5)(4,8)(6,7)(17,20,21,18)(19,24,23,22);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,13)(14,16)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=gh,e^2=hi,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 | #72232285, #72232288, #72300459, … |
| maximal normal subgroup of | #72232285, #72232288, #72300459, … |
| maximal quotient of | #72232307, #72232401, #72300529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8364041);