Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
5518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,16)(11,14,12,15)(18,22)(23,24);(1,2,5,4)(3,6,8,7)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22);(2,4)(6,7)(9,10)(11,15)(12,14)(13,16);(1,3,5,8)(2,6,4,7)(9,10,13,16)(11,14,12,15);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,20)(18,23,22,24);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72304313, #72304404 |
| maximal normal subgroup of | #72304313, #72304404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7691206);