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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1956 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)(7,11)(8,12)(9,10)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(6,10)(11,12)(14,19)(15,18)(16,23)(21,28)(24,25)(26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,18)(14,20,19,22)(16,23,24,25)(21,27,26,28);(2,4)(13,16)(14,21)(15,23)(17,24)(18,25)(19,26)(20,27)(22,28);(5,8,9,7)(6,11,10,12)(14,19)(20,22)(21,26)(27,28)Polycyclic presentation:
2^9|c^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,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, 7426843);