Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #25162 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,19,16)(15,20,25,26)(17,21,18,22)(23,27,28,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,24)(18,27)(19,25)(21,23)(22,28);(2,4)(6,9)(10,11)(13,16,19,14)(15,23,25,28)(17,21,18,22)(20,27,26,24);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,21,16,22)(15,24,25,27)(20,23,26,28);(15,24)(17,18)(20,23)(21,22)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578326);