Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,16,13,14,19,21)(11,15,17,22,18,20,23,24);(2,5)(3,8)(6,7)(10,14)(15,20);(2,6)(5,7);(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264399, #72264455, #72264458, … |
| maximal normal subgroup of | #72264399, #72264455, #72264458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389023);