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×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,19,15)(16,23,25,24)(17,27,21,18)(20,28,22,26);(2,4)(6,10)(11,12)(13,15,19,14)(16,24,25,23)(17,18,21,27)(20,26,22,28);(5,7,9,8)(6,11,10,12)(13,16,18,26,19,25,27,28)(14,20,21,23,15,22,17,24);(1,2,3,4)(13,17,18,14,19,21,27,15)(16,24,26,20,25,23,28,22);(5,8,9,7)(6,11,10,12)(16,25)(20,22)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=h,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, 7521687);