Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144641 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10391 |
| commutator, Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 380 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)(3,7)(4,14)(5,9,18,24)(6,10)(8,16,22,31)(11,25)(12,20,27,23)(13,21)(15,28)(17,29)(19,26,32,30);(1,3)(2,7)(4,15)(5,12,18,27)(6,13)(8,19,22,32)(9,23,24,20)(10,21)(11,17)(14,28)(16,30,31,26)(25,29);(1,4,6,17)(2,8)(3,11)(5,16,18,31)(7,19,21,32)(9,14)(10,22)(12,26)(13,15)(20,25,23,28)(24,29)(27,30)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=h,e^2=i,b^a=bd,c^a=ce,e^a=egi,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,f^c=fi,e^d=ei29 conjugacy classes — view the full character table.
| maximal subgroup of | #72235231, #72283440, #72283607, … |
| maximal normal subgroup of | #72235231, #72283440, #72283607, … |
| maximal quotient of | #72235243, #72235247, #72235375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51837);