Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 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)(15,18)(16,24)(17,19)(20,22)(21,26)(23,25)(27,28);(1,2)(3,4)(6,10)(11,12)(13,15,20,17)(14,18,22,19)(16,21,25,28)(23,27,24,26);(5,7,9,8)(6,11,10,12)(13,16,18,26,20,25,19,27)(14,21,17,24,22,28,15,23);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,14,20,15,18,22)(16,24,27,21,25,23,26,28);(5,8,9,7)(6,11,10,12)(16,25)(21,28)(23,24)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fgi,e^2=f,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=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472648);