Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139759 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9495 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,13,17,23)(4,18)(5,10,6,11)(7,12)(8,15)(9,19,26,21)(14,28)(16,24)(20,25)(22,31)(27,29,32,30);(1,3,5,16)(2,8,10,13)(4,14,19,29)(6,15,7,17)(9,22,18,27)(11,23,12,24)(20,31,21,30)(25,32,26,28);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,25)(12,26)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72251527, #72251567, #72258232, … |
| maximal normal subgroup of | #72251527, #72251567, #72258232, … |
| maximal quotient of | #72258408, #72262365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46956);