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 | |
7218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,23)(19,21,20,24);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,12,14,15)(18,23)(19,20);(2,7)(3,8)(9,12,13,15)(10,11,14,16)(17,19,22,20)(18,24,23,21);(1,4,5,6)(2,3,7,8)(9,10,13,14)(11,15,16,12)(17,20,22,19)(18,21,23,24);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,19)(20,23)(22,24);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72232278, #72232446, #72232449, … |
| maximal normal subgroup of | #72232278, #72232446, #72232449, … |
| maximal quotient of | #72232418, #72232435, #72232510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8502542);