Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,15,16,14,18,13,11)(12,17,22,23,21,24,20,19);(2,4)(10,16)(11,18)(12,20)(13,15)(19,23)(21,22);(5,6,7,8)(9,11,13,18,14,16,15,10)(12,19,20,24,21,23,22,17);(9,12,13,20,14,21,15,22)(10,17,11,19,18,24,16,23);(12,21)(17,24)(19,23)(20,22)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312807, #72312904, #72312918 |
| maximal normal subgroup of | #72312807, #72312904, #72312918 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372928);