Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1204 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)(9,10,13,17,14,16,12,22)(11,18,19,23,20,24,21,15);(1,2)(3,4)(9,11,12,21,14,20,13,19)(10,15,17,18,16,23,22,24);(2,4)(10,16)(11,19)(12,13)(15,18)(20,21)(23,24);(5,7,8,6)(11,20)(15,23)(18,24)(19,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=efh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321742 |
| maximal normal subgroup of | #72321742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266592);