Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 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)(7,11)(8,12)(9,10)(13,14,19,16)(15,23,26,21)(17,18,22,20)(24,25,28,27);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,21,16,23)(17,28,22,24)(18,25,20,27);(5,7,9,8)(6,11,10,12)(13,16,18,22,19,14,20,17)(15,24,27,23,26,28,25,21);(2,4)(13,17,18,16,19,22,20,14)(15,21,27,24,26,23,25,28);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,17,16,22)(15,25,26,27)(21,28,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471093);