Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #25081 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14)(15,21)(16,24)(17,22)(18,19)(20,25)(23,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(15,18)(16,23)(19,21)(20,28)(24,27)(25,26);(2,4)(13,15,17,18)(14,19,22,21)(16,23,25,26)(20,27,24,28);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,8,9,7)(6,11,12,10)(13,15,17,18)(14,21,22,19)(16,23,25,26)(20,28,24,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430120);