Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 512 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(21,22,28,34)(23,35,40,51)(24,31,42,48)(25,30,41,49)(26,33,27,32)(29,43,47,37)(36,45,52,46)(38,44,39,50);(1,2)(3,4)(8,17)(11,16)(13,20)(15,18)(21,23,28,40)(22,29,34,47)(24,37,42,43)(25,36,41,52)(26,39,27,38)(30,35,49,51)(31,44,48,50)(32,46,33,45);(2,4)(5,7)(6,10)(8,18)(9,14)(11,20)(12,19)(13,16)(15,17)(21,24,27,41,28,42,26,25)(22,30,33,48,34,49,32,31)(23,36,39,43,40,52,38,37)(29,44,46,51,47,50,45,35)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=g,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50561);