Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139598 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #2021 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 535 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,22,26)(19,27,29,46)(20,32,28,25)(21,34)(23,38,41,30)(24,37)(31,44)(33,43)(35,45)(36,42)(39,47)(40,48);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,28,38,22,29,20,30)(18,23,32,46,26,41,25,27)(21,35,36,47,37,31,33,48)(24,40,42,44,34,39,43,45);(3,11)(7,15)(10,16)(12,13)(18,24)(19,29)(23,39)(25,43)(26,34)(27,44)(30,38)(31,35)(32,42)(40,41)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72284217, #72284240, #72284275, … |
| maximal normal subgroup of | #72284217, #72284240, #72284275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46842);