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 | |
6209 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,23)(22,24);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,14)(13,15);(1,2)(3,8)(4,6)(5,7)(9,11,13,12)(10,14,15,16)(17,19,22,21)(18,23,24,20);(9,10)(11,16)(12,14)(13,15)(17,20,22,23)(18,19,24,21);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,23,24,20);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233547, #72233690, #72304347, … |
| maximal normal subgroup of | #72233547, #72233690, #72304347, … |
| maximal quotient of | #72303653, #72303692, #72304069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592524);