Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,20,15)(16,25,28,22)(17,21,27,24)(18,23,19,26);(1,2)(3,4)(6,10)(11,12)(13,15,20,14)(16,22,28,25)(17,24,27,21)(18,26,19,23);(5,7,9,8)(6,11,10,12)(13,16,18,27,20,28,19,17)(14,21,26,22,15,24,23,25);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,28,20,27,18,16)(14,22,23,21,15,25,26,24);(6,10)(11,12)(13,18,20,19)(14,23,15,26)(16,27,28,17)(21,25,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,e^2=i,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=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469056);