Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144653 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10392 |
| commutator, Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #3486 | C22 |
| LCS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,13,19,10)(3,14,15,8,4,18,17,26)(6,11,22,30,23,31,21,24)(9,27,28,20,29,32,25,16);(1,3,5,15,7,4,19,17)(2,8,10,14,13,26,12,18)(6,20,22,32,23,16,21,27)(9,11,25,24,29,31,28,30);(1,4)(2,9)(3,7)(5,17)(6,16)(8,24)(10,28)(11,14)(12,25)(13,29)(15,19)(18,31)(20,23)(21,32)(22,27)(26,30)Polycyclic presentation:
2^9|a^2=dg,b^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fgi,h^b=hi,d^c=dg,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^e=fi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209704 |
| maximal normal subgroup of | #72209704 |
| maximal quotient of | #72210062, #72210063, #72210064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51849);