Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34738 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3438 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)(7,8)(9,10,14,17,13,16,22,24)(11,12,19,21,18,20,15,23);(1,2)(3,4)(9,11,14,19,13,18,22,15)(10,12,17,21,16,20,24,23);(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(2,4);(10,16)(12,20)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,e^c=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72315645, #72315668, #72316079 |
| maximal normal subgroup of | #72315645, #72315668, #72316079 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288548);