Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462214 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16422 |
| 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 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,14)(4,9)(5,19,22,29)(6,11,24,13)(8,25)(10,21,28,23)(15,31)(16,18)(17,32)(20,26)(27,30);(1,3)(2,8,12,18)(4,11,14,13)(5,15)(6,16)(7,9)(10,20,28,31)(17,19,30,29)(21,32)(22,26)(23,27)(24,25);(1,4,6,18,7,14,24,8)(2,9,13,25,12,3,11,16)(5,20,21,17,22,31,23,30)(10,27,29,15,28,32,19,26);(1,5,6,21,7,22,24,23)(2,10,11,19,12,28,13,29)(3,15,16,32,9,26,25,27)(4,17,8,20,14,30,18,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=ef,d^2=e,e^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,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369410);