Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,16)(12,21)(13,18)(14,17)(15,20)(19,24)(22,23);(6,7)(11,19)(12,14)(15,23)(16,24)(17,21)(20,22);(1,2,3,4)(5,6,8,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,18)(17,24)(21,23);(5,7,8,6)(9,11)(10,16)(12,19)(13,20)(14,22)(15,18)(17,23)(21,24);(9,12,13,14)(10,17,18,21)(11,19,20,22)(15,24,16,23)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312922, #72313036, #72313050 |
| maximal normal subgroup of | #72312922, #72313036, #72313050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351379);