Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471189 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16773 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,14,6,13)(3,15,18,9,20,28,19,29)(4,21,23,10,24,11,5,25)(16,31,17,30,22,26,32,27);(1,3,6,19,7,20,8,18)(2,9,13,15,14,29,12,28)(4,22,23,32,24,16,5,17)(10,26,11,27,25,31,21,30);(1,4,7,24)(2,10,14,25)(3,16)(5,8,23,6)(9,26)(11,13,21,12)(15,30)(17,18)(19,32)(20,22)(27,28)(29,31);(1,5,6,4,7,23,8,24)(2,11,12,10,14,21,13,25)(3,17,19,22,20,32,18,16)(9,27,28,26,29,30,15,31)Polycyclic presentation:
2^9|a^2=eg,b^2=h,c^2=ef,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378385);