Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456890 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16287 |
| 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 | |
1165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,3,6,11)(4,15,17,13,18,28,23,9)(5,10,21,27,22,14,20,26)(16,25,19,29,31,30,32,24);(1,3,8,2)(4,9)(5,14,22,10)(6,11,7,12)(13,18)(15,23)(16,24)(17,28)(19,30)(20,26,21,27)(25,32)(29,31);(1,4,6,17,8,18,7,23)(2,9,11,15,3,13,12,28)(5,19,20,31,22,32,21,16)(10,25,26,29,14,30,27,24);(1,5)(2,10)(3,14)(4,16)(6,21)(7,20)(8,22)(9,24)(11,27)(12,26)(13,29)(15,30)(17,32)(18,31)(19,23)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237067, #72237078, #72237082, … |
| maximal normal subgroup of | #72237067, #72237078, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364087);