Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(19,23)(22,24);(9,11)(10,14)(12,15)(13,16)(18,21)(22,24);(1,2,4,3)(5,6,8,7)(9,10,13,12)(11,14,16,15)(17,18,20,21)(19,22,23,24);(5,6,8,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24);(1,3,4,2)(5,7,8,6)(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|c^2=gh,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72297130, #72297168, #72297184 |
| maximal normal subgroup of | #72297130, #72297168, #72297184 |
| maximal quotient of | #72298210 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1317029);