Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2970 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)(9,10,13,12,14,17,15,18)(11,19,21,24,22,23,20,16);(5,6)(7,8)(9,11)(10,16)(12,23)(13,20)(14,22)(15,21)(17,24)(18,19);(1,2,3,4)(6,8)(9,10,15,18,14,17,13,12)(11,19,20,16,22,23,21,24);(6,8)(9,12,13,17,14,18,15,10)(11,16,21,19,22,24,20,23);(9,13,14,15)(10,12,17,18)(11,20,22,21)(16,23,24,19)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307586, #72307595, #72307626, … |
| maximal normal subgroup of | #72307586, #72307595, #72307626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351540);