Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1434 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
866 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 162 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)(11,19,20,23)(12,15)(13,17)(14,18)(16,22,24,21);(10,15,18,17)(11,20)(16,19,24,23)(21,22);(1,3)(2,5)(4,8)(6,7)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280575, C2×#108573 |
| maximal normal subgroup of | #72280575, C2×#108573 |
| maximal quotient of | C2×#108573 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15769);