Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16425 |
| 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 | |
694 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)(3,15,10,4)(5,20)(6,12)(7,14)(8,13)(9,17,24,18)(11,22)(16,21,26,32)(19,31,30,27)(23,28)(25,29);(1,3,8,18,7,10,6,17)(2,9,12,15,14,24,13,4)(5,16,25,27,23,26,22,31)(11,21,28,30,29,32,20,19);(1,4,6,24,7,15,8,9)(2,10,13,18,14,3,12,17)(5,21,22,19,23,32,25,30)(11,27,20,16,29,31,28,26);(1,5,6,22,7,23,8,25)(2,11,12,28,14,29,13,20)(3,16,17,31,10,26,18,27)(4,19,9,21,15,30,24,32)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=efi,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369485);