Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,15,18)(11,13,17,14)(12,20,23,16)(19,24,21,22);(5,6)(7,8)(9,11,14,10,15,17,13,18)(12,21,24,20,23,19,22,16);(1,2,3,4)(6,8)(9,12,15,23)(10,16,18,20)(11,19,17,21)(13,22,14,24);(6,8)(9,13,15,14)(10,17,18,11)(12,22,23,24)(16,21,20,19);(12,23)(16,20)(19,21)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305329, #72305330, #72305339, … |
| maximal normal subgroup of | #72305329, #72305330, #72305339, … |
| maximal quotient of | #72305623, #72305644, #72305901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350309);