Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×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×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,9,11,10,7,8,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(1,3,4,2)(5,8,10,9)(6,11,7,12)(15,21)(18,24)(22,27)(25,28);(5,8,10,9)(6,11,7,12)(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,c^b=ci,e^b=eh,f^b=fh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90595);