Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,17)(11,19,21,15)(12,22,20,16)(14,18,23,24);(1,2,3,4)(9,11)(10,15)(12,23)(13,21)(14,20)(16,24)(17,19)(18,22);(6,8)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24);(9,12)(10,16)(11,14)(13,20)(15,18)(17,22)(19,24)(21,23);(11,12)(14,23)(15,16)(18,24)(19,22)(20,21)Polycyclic presentation:
2^9|a^2=i,b^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285305 |
| maximal normal subgroup of | #72285305 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481081);