Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18,20,22)(19,24,21,23);(2,5)(6,8)(11,12)(14,15)(18,22)(23,24);(9,11,13,12)(10,14,16,15)(18,23)(19,21)(22,24);(9,12,13,11)(10,15,16,14);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(17,19,20,21)(18,24,22,23)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284713, #72284745, #72285065, … |
| maximal normal subgroup of | #72284713, #72284745, #72285065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6395449);