Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #24362 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2051 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)(7,10,11,12)(13,14,18,19)(15,16,20,22)(17,21,25,26)(23,24,27,28);(6,9)(10,12)(13,15)(14,16)(17,24)(18,20)(19,22)(21,27)(23,26)(25,28);(2,4)(14,19)(15,22)(16,20)(17,25)(23,24)(27,28);(5,7,8,11)(6,10,9,12)(13,16,18,22)(14,20,19,15)(17,23,25,27)(21,24,26,28);(13,17,18,25)(14,21,19,26)(15,23,20,27)(16,24,22,28)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483353);