Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | #390 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 56 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,28,30,23)(20,24,32,38)(21,26,34,25)(22,27)(29,42,44,35)(31,39,46,36)(33,37)(40,47)(41,45,48,43);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,21,30,22,31,34,46)(18,23,26,28,27,36,25,39)(20,29,33,45,32,43,47,44)(24,35,37,41,38,48,40,42);(2,7)(6,14)(8,15)(9,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,35)(25,40)(26,37)(27,38)(28,41)(30,45)(31,43)(34,47)(36,48)(39,42)(44,46)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gh,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efghi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ehi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57823);