Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,28)(23,24,25,27);(1,2,3,4)(13,15,16,19)(14,17,18,20)(21,23,26,25)(22,27,28,24);(2,4)(6,9)(10,12)(21,24,26,27)(22,23,28,25);(14,18)(17,20)(21,25,26,23)(22,24,28,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518886);