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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2140 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)(8,9)(10,12)(13,14,20,24)(15,17,23,19)(16,27,22,21)(18,28,25,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,14,20,23,17,24)(16,18,26,27,22,25,28,21);(6,9)(10,11)(13,16,20,22)(14,21,24,27)(15,25,23,18)(17,28,19,26);(2,4)(5,7,8,12)(6,10,9,11)(13,17,20,19)(14,15,24,23)(16,26,22,28)(18,27,25,21);(13,18,17,21,20,25,19,27)(14,22,15,26,24,16,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458408);