Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16420 |
| 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 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,13,6,11)(3,4,15,18,16,19,25,9)(5,20,24,10,23,28,22,29)(14,21,27,32,31,26,30,17);(1,3,7,16)(2,9)(4,11)(5,14,23,31)(6,15,8,25)(10,26)(12,19)(13,18)(17,20)(21,29)(22,27,24,30)(28,32);(1,4,8,18,7,19,6,9)(2,3,11,25,13,16,12,15)(5,21,24,32,23,26,22,17)(10,27,20,14,29,30,28,31);(1,5,6,22,7,23,8,24)(2,10,11,20,13,29,12,28)(3,14,15,27,16,31,25,30)(4,17,18,21,19,32,9,26)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=fi,d^2=e,e^2=i,f^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,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369355);