Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1526 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 146 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)(4,6)(5,8)(9,10,12,16,13,17,20,24)(11,15,18,23,19,14,21,22);(1,3)(2,5,6,7)(4,8)(10,14,17,15)(16,23,24,22);(12,20)(16,24)(18,21)(22,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,d^b=dh,e^b=egi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213489, #72215527, #72215559, … |
| maximal normal subgroup of | #72213489, #72215527, #72215559, … |
| maximal quotient of | C2×#128059, #72261314, #72262212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35255);