Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21);(6,8)(9,11,13,16)(10,12,14,15)(17,19,20,24)(18,21,22,23);(5,6)(7,8)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#557552, #72293946, #72297178, … |
| maximal normal subgroup of | C2×#557552, #72293946, #72297178, … |
| maximal quotient of | #72293743, C2×#557552, #72298219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 464748);