Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3334 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)(3,4)(5,6,8,7)(10,15)(12,13)(16,17)(18,23)(19,24)(20,22);(5,7,8,6)(9,10)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24);(2,4)(9,11)(10,16)(12,22)(13,20)(14,21)(15,17)(18,19)(23,24);(9,11)(10,17)(12,20)(13,22)(14,21)(15,16)(18,24)(19,23);(9,12,14,13)(10,15,18,23)(11,20,21,22)(16,24,19,17)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ci,d^b=df,f^b=fi,e^c=ei,f^d=fi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284435, #72284921 |
| maximal normal subgroup of | #72284435, #72284921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288634);