Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1220 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8,5,3,4,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,3,5,2)(4,8)(6,7)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24);(4,6)(7,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72227781, #72227862, #72233244, … |
| maximal normal subgroup of | #72227781, #72227862, #72233244, … |
| maximal quotient of | #72227780, #72251735, #72317977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47024);