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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1862 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)(13,14,17,19)(15,20,23,26)(16,24,18,21)(22,28,25,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(14,19)(15,22)(16,18)(20,27)(23,25)(26,28);(2,4);(2,4)(5,7,9,8)(6,11,12,10)(13,15)(14,20)(16,22)(17,23)(18,25)(19,26)(21,27)(24,28);(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,21,19,24)(15,22,23,25)(20,27,26,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=fi,e^2=f,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, 7430125);