Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| 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 | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1406 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)(7,14)(8,12)(9,18)(10,13)(11,15)(16,19)(17,20)(21,22)(23,26,30,34)(24,27)(25,28)(29,33,31,35)(32,36);(5,7,9,15,10,16,8,17)(6,11,12,14,13,20,18,19)(21,23)(22,26,28,34)(24,31)(25,30)(27,35,36,33)(29,32);(2,4)(24,32)(27,36)(29,31)(33,35);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,20,14)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103650);