Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,18,16,11,15,12)(13,20,22,17,24,21,23,19);(1,2)(3,4)(6,7)(9,11)(10,16)(12,14)(13,21)(15,18)(17,23)(19,22)(20,24);(2,4)(5,6,8,7)(9,12,14,11,16,18,15,10)(13,20,22,19,24,21,23,17);(5,7,8,6)(9,13,15,23,16,24,14,22)(10,17,18,21,11,19,12,20)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327715);