Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 52 points:
(2,4)(5,6,9,13,10,14,8,12)(7,15,17,11,18,19,16,20)(21,22,26,33,28,35,25,30,29,36,24,32,27,34,44,47)(23,37,40,31,42,46,45,50,43,52,38,51,41,48,39,49);(1,2,3,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,15)(14,20)(22,30)(24,25)(26,44)(27,28)(31,46)(32,34)(33,35)(36,47)(37,50)(38,45)(39,40)(41,42)(48,51)(49,52);(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,19)(14,20)(21,23,29,43)(22,31,36,51)(24,39,26,45)(25,40,44,38)(27,42,28,41)(30,46,47,48)(32,37,33,52)(34,50,35,49)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44625);