Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144399 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10337 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23897 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
798 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,11,14,7)(4,8)(5,10)(6,9)(12,29,30,21)(13,22,18,27)(15,32,31,23)(16,28)(17,24)(19,25)(20,26);(1,3)(2,7)(4,15)(5,18)(6,14)(8,23)(9,11)(10,27)(12,16)(13,20)(17,31)(19,30)(21,25)(22,26)(24,32)(28,29);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,28)(11,29)(13,31)(14,30)(15,18)(16,20)(22,32)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgh,c^b=cf,e^b=eh,f^b=fi,g^b=gi,d^c=dg,f^c=fi,h^c=hi,f^d=fi,f^e=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72237635, #72237640, #72255685, … |
| maximal normal subgroup of | #72237635, #72237640, #72255685, … |
| maximal quotient of | #72251664, #72251669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51595);