Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486091 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #12094 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
820 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,40)(28,39)(29,35)(30,41)(31,36)(32,42)(33,38)(34,37)(43,46)(44,47)(45,48);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,23)(19,27)(24,38)(25,36)(26,37)(28,30)(29,33)(35,40)(39,47)(41,46)(42,48)(43,44);(17,19,22,27)(18,24,26,40)(20,30,32,28)(21,33,34,29)(23,35,37,38)(25,41,42,39)(31,43,45,44)(36,47,48,46);(2,7)(3,10)(8,12)(11,13)(17,20,22,32)(18,25,26,42)(19,28,27,30)(21,31,34,45)(23,36,37,48)(24,39,40,41)(29,43,33,44)(35,46,38,47)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393302);