Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144633 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10390 |
| 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 | |
505 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 505 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)(5,9,18,26)(6,10)(8,15,23,17)(11,27)(12,20,31,24)(13,21)(16,25)(19,28,22,30)(29,32);(1,3)(2,7)(4,11)(5,12,18,31)(6,13)(8,22,23,19)(9,24,26,20)(10,21)(14,27)(15,28,17,30)(16,29)(25,32);(1,4,6,16)(2,8,10,23)(3,11,13,29)(5,17,18,15)(7,19,21,22)(9,25,26,14)(12,30,31,28)(20,32,24,27)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=i,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,e^c=ei,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72258232, #72258235, #72268770, … |
| maximal normal subgroup of | #72258232, #72258235, #72268770, … |
| maximal quotient of | #72258399 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51830);