Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #391 | C23 |
| LCS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ G | |
| LECS | G ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 38 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:
(2,6)(3,4)(7,8)(10,14)(11,18)(15,23)(16,24)(17,22)(19,20);(1,2)(3,6)(4,8)(5,7)(10,15)(11,19)(12,21)(14,22)(18,20)(23,24);(1,3,5,4)(2,6,7,8)(9,10,12,16,13,15,21,17)(11,14,20,22,18,23,19,24)Polycyclic presentation:
2^9|c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,c^b=ch,d^b=df,e^b=eghi,g^b=gi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72228064 |
| maximal normal subgroup of | #72228064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58546);