Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,20,24)(15,19,23,18)(16,25,28,17)(21,22,26,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,22)(18,24)(20,23)(21,28)(25,27);(6,9)(10,11)(13,16,19,27,20,28,18,22)(14,21,15,25,24,26,23,17);(5,7,8,12)(6,10,9,11)(13,17,19,21,20,25,18,26)(14,22,15,16,24,27,23,28);(1,3,4,2)(13,18,20,19)(14,15,24,23)(16,27,28,22)(17,26,25,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=g,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503988);