Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | C2×#449 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,32)(24,31)(25,30)(26,29)(27,34)(28,33)(35,36);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,34)(22,27,30,32)(24,35,26,33)(28,29,36,31);(7,14)(10,13)(15,20)(17,19)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,32)(30,36);(6,11)(10,13)(12,18)(17,19)(21,24,25,26)(22,29,30,31)(23,33,34,35)(27,28,32,36);(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=hi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283175);