Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,32)(31,34)(35,36);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23,24,30,25,29,31,35)(22,26,28,32,27,33,34,36);(1,3,4,2)(7,14)(10,18)(13,19)(15,20);(7,14)(10,18)(13,19)(15,20)(22,27)(23,29)(28,34)(30,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,g^2=i,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,d^b=dhi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89453);