Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1599 conjugacy classes of subgroups; only the 494 normal subgroups are shown.
This group has 1599 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,9,13,16)(5,14,15,8)(7,11,10,12)(17,18)(19,23)(20,25)(21,24)(22,26)(27,29)(28,32)(30,31);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,14)(12,15)(17,19,22,28)(18,23,26,32)(20,29,30,21)(24,25,27,31);(1,4,6,2)(3,10,13,7)(5,14,15,8)(9,12,16,11)(18,24)(19,28)(21,29)(23,31)(25,32)(26,27);(3,11)(5,15)(7,16)(8,14)(9,10)(12,13)(17,20)(18,25)(19,29)(21,28)(22,30)(23,27)(24,32)(26,31);(2,4)(7,10)(8,14)(9,16)(18,26)(19,28)(20,30)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276368);