Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1101 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,5,8,6,3,4,7)(9,10)(11,17,19,23)(12,20,22,15)(13,16)(14,18)(21,24);(1,3,6,2)(4,7,5,8)(9,11,13,19)(10,14,16,24)(12,21,22,18)(15,23,20,17);(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fg,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72209918, #72227596, #72247303, … |
| maximal normal subgroup of | #72227596, #72247303, #72258062, … |
| maximal quotient of | #72209913, #72227593 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38639);