Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7)(9,10,12,11)(22,25)(23,27);(2,3)(9,10,12,11)(14,17)(18,20)(22,25)(26,28);(5,7,8,6)(9,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=efg,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61186);