Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,2)(3,5)(4,6)(7,8)(9,11,12,20,13,19,21,18)(10,14,16,24,15,22,23,17);(1,3)(2,5)(4,8)(6,7);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281137, #72281184, #72312114, … |
| maximal normal subgroup of | #72281137, #72281184, #72312114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101853);