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 | |
7688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,7)(3,4)(10,11)(14,15);(1,2,6,7)(3,8,4,5)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,23)(21,24);(1,3,6,4)(2,8,7,5)(17,19)(18,21)(20,24)(22,23);(1,3,6,4)(2,5,7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(1,4,6,3)(2,5,7,8);(1,5)(2,3)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233515, #72233542, #72233598, … |
| maximal normal subgroup of | #72233515, #72233542, #72233598, … |
| maximal quotient of | #72261116, #72262118, #72297717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329358);