Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139718 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #15763 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 931 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,7,8,5)(4,6)(9,10,14,16,13,17,23,20)(11,18,15,24,21,12,22,19);(1,2)(3,5)(4,6)(7,8)(9,11,19,10,13,21,24,17)(12,20,23,22,18,16,14,15);(1,3,4,8)(2,5,6,7)(12,18)(14,23)(15,22)(16,20)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=hi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eh,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72235017, #72235018, #72235020, … |
| maximal normal subgroup of | #72235017, #72235018, #72235020, … |
| maximal quotient of | #72238534 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46993);