Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10,14,17)(8,11)(9,12)(13,16,19,20)(15,18)(21,22,25,28,26,29,24,32)(23,27,30,35,31,36,33,34);(5,7,9,14)(6,10)(8,13,15,19)(11,16)(12,17)(18,20)(21,23,24,30,26,31,25,33)(22,27,28,34,29,36,32,35);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20);(8,15)(11,18)(13,19)(16,20)(21,24,26,25)(22,28,29,32)(23,30,31,33)(27,34,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,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, 103267);