Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | C22×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,13,19,15,20,14,18)(11,21,22,16,12,24,23,17);(5,6)(7,8)(9,11,13,22,15,12,14,23)(10,16,18,21,20,17,19,24);(1,2,3,4)(6,8)(9,10,15,20)(11,16,12,17)(13,18,14,19)(21,23,24,22);(9,12,14,22,15,11,13,23)(10,17,19,21,20,16,18,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fhi,d^2=ei,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 | #72226660, #72226672, #72226677 |
| maximal normal subgroup of | #72226660, #72226672, #72226677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281633);