Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485047 |
| 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 | #3490 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 120 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)(6,10)(8,14)(11,16)(13,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,40)(29,41)(31,37)(32,38)(33,46)(34,39)(35,43)(36,44)(42,48)(45,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,27)(26,36)(31,43)(32,42)(33,45)(34,44)(37,40)(38,47)(39,41)(46,48);(17,20,22,32)(18,24,26,38)(19,28,29,42)(21,33,34,31)(23,35,36,47)(25,37,39,46)(27,40,41,48)(30,45,44,43);(2,7)(6,15)(8,12)(9,16)(19,29)(23,36)(27,41)(28,42)(30,44)(35,47)(40,48)(43,45)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392282);