Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422306 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15304 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,3,7,14)(4,15,19,12,20,16,18,9)(5,21,22,10,24,26,23,13)(17,27,31,29,32,25,30,28);(1,3)(2,8)(4,16,20,15)(5,21)(6,11)(7,14)(9,19,12,18)(10,23)(13,22)(17,27,32,25)(24,26)(28,30,29,31);(1,4)(2,9)(3,12)(5,17)(6,18)(7,19)(8,20)(10,25)(11,16)(13,27)(14,15)(21,28)(22,30)(23,31)(24,32)(26,29);(1,5,6,22,8,24,7,23)(2,10,11,26,3,13,14,21)(4,17,18,30,20,32,19,31)(9,25,16,29,12,27,15,28)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318894, #72318916, #72318991, … |
| maximal normal subgroup of | #72318894, #72318916, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329503);