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 | |
6469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,13,12)(11,15,16,14)(18,20)(22,24);(1,2,5,4)(3,7,8,6)(9,11,13,16)(10,14,12,15)(19,23)(22,24);(2,4)(6,7)(10,12)(14,15)(17,18,21,20)(19,22,23,24);(9,11)(10,14)(12,15)(13,16)(17,19,21,23)(18,22,20,24);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297124, #72297317 |
| maximal normal subgroup of | #72297124, #72297317 |
| maximal quotient of | #72297163, #72297377, #72297527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161872);