Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162118 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2847 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1415 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)(3,6)(4,5)(7,8)(10,13)(11,15)(14,16)(17,18)(19,21,23,24)(20,22);(9,10,11,13,12,14,15,16)(17,19)(18,21,22,24)(20,23);(2,5)(6,8)(18,22)(21,24);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=fi,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309208, #72309244, #72309302, … |
| maximal normal subgroup of | #72309208, #72309244, #72309302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 115046);