Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
872 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)(3,4)(5,6,10,7,9,8,11,12)(13,14,17,16,18,20,26,25)(15,19,23,24,21,27,28,22);(2,4)(5,7,10,8,9,12,11,6)(13,15)(14,19)(16,24)(17,23)(18,21)(20,27)(22,25)(26,28);(5,7,10,8,9,12,11,6)(14,20)(15,21)(16,25)(23,28);(5,8,10,12,9,6,11,7)(13,16,17,20,18,25,26,14)(15,22,23,19,21,24,28,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=dhi,c^b=chi,d^b=dh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91314);