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 | |
1888 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)(13,14,18,17)(15,19,23,22)(16,20,25,26)(21,27,28,24);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,23)(14,19,17,22)(16,24,25,27)(20,21,26,28);(2,4);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,20)(15,21)(17,26)(18,25)(19,27)(22,24)(23,28);(5,8,9,7)(6,11,12,10)(13,14,18,17)(15,22,23,19)(16,20,25,26)(21,24,28,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,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, 7430140);