Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18,22,23,21,20,19,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,22)(18,23,20,24);(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,24,22,23);(3,8)(5,7)(10,14)(11,16)(17,19,21,22)(18,23,20,24);(2,6)(5,7)(10,14)(13,15)(17,19,21,22)(18,23,20,24)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=hi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7276051);