Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,25)(24,26)(27,28);(2,4)(6,9)(10,12)(21,23,24,27)(22,25,26,28);(1,2,3,4)(14,17)(18,20);(13,15,16,19)(14,18,17,20)(22,26)(25,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 515392);