Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10,15,19,16,11,14,18)(12,21,13,17,23,24,22,20);(5,6,7,8)(9,11,16,10)(12,20,23,17)(13,24,22,21)(14,18,15,19);(9,12,15,13,16,23,14,22)(10,17,18,21,11,20,19,24);(9,13,14,12,16,22,15,23)(10,17,18,21,11,20,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72226837, #72226841, #72226843, … |
| maximal normal subgroup of | #72226837, #72226841, #72226843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263537);