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 | |
6098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,7)(3,8,4,5)(17,18,21,19)(20,23,24,22);(1,3,6,4)(2,5,7,8)(9,10,13,12)(11,14,15,16)(18,19)(22,23);(2,7)(5,8)(10,12)(14,16)(17,19,21,18)(20,23,24,22);(2,7)(3,4)(11,15)(14,16)(17,20,21,24)(18,22,19,23);(1,4,6,3)(2,8,7,5)(9,11,13,15)(10,14,12,16);(1,5,6,8)(2,3,7,4)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297317, #72297328, C2×#8341810, … |
| maximal normal subgroup of | #72297317, #72297328, C2×#8341810, … |
| maximal quotient of | #72297341, #72297354, #72297507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8249006);