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 | |
6095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,15,16,14)(18,20)(22,24);(2,6)(3,8)(10,13)(14,15)(19,23)(22,24);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24);(1,3)(2,7)(4,6)(5,8)(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,24,23,22);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72297213, #72297326, #72311133, … |
| maximal normal subgroup of | #72297213, #72297326, #72311133, … |
| maximal quotient of | #72297260, #72297407, #72297743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412507);