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 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1157 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1157 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,8,11,16)(4,14,13,7)(5,15,12,9)(17,18)(19,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31);(1,3,6,11)(2,7,10,14)(4,12,13,5)(8,9,16,15)(17,19,22,18)(20,29,31,28)(21,27,23,25)(24,30,26,32);(3,11)(5,12)(7,14)(9,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,29)(27,28);(1,4)(2,8)(3,12)(5,11)(6,13)(7,9)(10,16)(14,15)(17,21,22,23)(18,25,19,27)(20,30,31,32)(24,29,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,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, 151967);