Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,17)(12,20,22,15)(13,19)(14,16)(18,21,24,23);(1,3,5,7,6,2,4,8)(9,11,17,10,14,19,13,16)(12,18,21,15,22,24,23,20);(9,12)(10,15)(11,18)(13,23)(14,22)(16,20)(17,21)(19,24);(1,4,6,5)(2,7,3,8)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,f^b=fh,e^c=eh,g^c=gh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281514, #72281669, #72302217 |
| maximal normal subgroup of | #72281514, #72281669, #72302217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103432);