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 | |
6085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,13)(12,16)(14,15)(17,18,21,22)(19,20,23,24);(1,2)(3,8)(4,5)(6,7)(10,15)(11,16)(18,22)(19,23);(2,7)(3,4)(9,11,14,16)(10,12,15,13)(17,19)(18,20)(21,23)(22,24);(1,3,6,4)(2,8,7,5)(9,12,14,13)(10,11,15,16)(17,20)(18,19)(21,24)(22,23);(1,4,6,3)(2,5,7,8)(9,10)(11,12)(13,16)(14,15);(1,5)(2,3)(4,7)(6,8)(9,13,14,12)(10,16,15,11)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258018, #72297127, #72319798, … |
| maximal normal subgroup of | #72258018, #72297127, #72319798, … |
| maximal quotient of | #72297160, #72297198, #72297937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706678);