Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10,14,16)(8,12)(9,11)(13,17,19,20)(15,18)(21,22,25,28,24,27,31,33)(23,26,30,35,29,34,32,36);(5,7)(6,10,11,16)(8,13)(9,14)(12,17,18,20)(15,19)(21,23,25,32,24,29,31,30)(22,26,33,35,27,34,28,36);(1,2,3,4)(6,11)(8,15)(10,16)(13,19)(22,27)(26,34)(28,33)(35,36);(5,8,9,15)(6,12,11,18)(7,13,14,19)(10,17,16,20)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=dhi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111448);