Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,35)(34,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,24,31)(22,26,28,30)(25,32,33,36)(27,34,29,35);(7,14)(10,13)(15,20)(17,19)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(6,11)(10,13)(12,18)(17,19)(22,28)(26,30)(27,29)(34,35);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281853);