Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,20,16,19,15,12)(11,18,13,17,22,24,21,23);(1,2,4,3)(5,6)(7,8)(9,11,14,13,16,22,15,21)(10,17,12,18,19,23,20,24);(1,3,4,2)(6,8)(9,12,14,19,16,20,15,10)(11,18,13,23,22,24,21,17);(9,13,14,22,16,21,15,11)(10,18,12,23,19,24,20,17)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320933, #72321026, #72321053, … |
| maximal normal subgroup of | #72320933, #72321026, #72321053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319925);