Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1168 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)(9,10,15,18,14,20,13,19)(11,21,24,22,23,17,12,16);(2,4)(5,7,8,6)(9,11,13,12,14,23,15,24)(10,16,18,21,20,22,19,17);(1,2,3,4)(9,10,14,20)(11,22,23,16)(12,17,24,21)(13,18,15,19);(9,12,13,23,14,24,15,11)(10,17,18,16,20,21,19,22)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321134, #72321156, #72321262, … |
| maximal normal subgroup of | #72321134, #72321156, #72321262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275860);