Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16557 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,12,5,11)(3,13,14,24,17,22,16,8)(4,18,19,26,20,25,21,9)(15,23,28,27,29,32,30,31);(1,3,5,16,6,17,7,14)(2,8,10,13,12,24,11,22)(4,15,21,30,20,29,19,28)(9,23,18,27,26,32,25,31);(3,14)(4,19)(5,7)(8,22)(9,25)(10,11)(13,24)(15,29)(16,17)(18,26)(20,21)(23,32);(1,4)(2,9)(3,15)(5,21)(6,20)(7,19)(8,23)(10,18)(11,25)(12,26)(13,27)(14,28)(16,30)(17,29)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374246);