Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,20,16)(15,21,26,27)(17,18,22,19)(23,25,28,24);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,27)(17,28)(18,24)(19,25)(20,26)(22,23);(13,16,18,22,20,14,19,17)(15,23,25,27,26,28,24,21);(2,4)(13,17,18,16,20,22,19,14)(15,21,25,23,26,27,24,28);(5,8,9,7)(6,11,12,10)(13,18,20,19)(14,22,16,17)(15,24,26,25)(21,23,27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=g,d^2=g,e^2=fi,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=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470843);