Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(4,5)(7,8)(9,10)(11,14,18,20)(12,16)(13,15)(17,21,23,24)(19,22);(1,2,5,7,6,8,4,3)(9,11,13,18)(10,14)(12,17,19,23)(15,20)(16,21)(22,24);(1,3,4,8,6,7,5,2)(10,15)(14,20)(16,22)(21,24);(9,12,13,19)(10,16,15,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72342997, D8×#410, #72343008, … |
| maximal normal subgroup of | #72342997, D8×#410, #72343008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112937);