Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244655 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3481 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
989 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 989 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,4)(3,11,14,9)(5,8,16,12)(6,15,13,10)(17,18)(19,20)(21,25)(22,26)(23,27)(24,31)(28,30)(29,32);(1,3)(2,8)(4,12)(5,13)(6,16)(7,14)(9,10)(11,15)(17,19)(18,20)(21,28)(22,24)(23,29)(25,30)(26,31)(27,32);(1,4,7,2)(3,12,14,8)(5,11,16,9)(6,10,13,15)(17,20)(18,24)(19,25)(21,30)(22,31)(23,32)(26,29)(27,28);(1,5,7,16)(2,9,4,11)(3,13,14,6)(8,10,12,15)(17,21,23,22)(18,25,27,26)(19,28,29,24)(20,30,32,31)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=eh,f^d=fh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151958);