Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(23,27)(26,28);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(14,18)(19,20)(22,25)(26,28);(13,15,17,16)(14,19,18,20)(21,23,24,27)(22,26,25,28);(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=h,e^2=hi,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, 515974);