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 | #24218 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 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)(16,24,27,28)(17,18,22,19)(21,25,23,26);(1,2)(3,4)(6,10)(11,12)(13,15,20,14)(16,24,27,28)(17,19,22,18)(21,25,23,26);(5,7,9,8)(6,11,10,12)(13,16,19,26,20,27,18,25)(14,21,22,24,15,23,17,28);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,14,20,22,18,15)(16,24,26,23,27,28,25,21);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,22,15,17)(16,25,27,26)(21,24,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472425);