Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,16,22,17,21,18,19)(15,23,25,27,26,20,24,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(15,24)(16,18)(20,23)(21,22)(25,26);(2,4)(6,10)(9,12)(13,15,16,25,17,26,18,24)(14,20,22,28,21,23,19,27);(2,4)(13,14,16,22,17,21,18,19)(15,20,25,28,26,23,24,27);(14,21)(19,22)(20,23)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348967);