Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128836 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1762 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,6)(3,4,7,8)(10,15,17,23)(11,19)(12,13)(16,20,18,24)(21,22);(1,3,4,6,5,7,8,2)(9,10)(11,20)(12,18)(13,16)(14,17)(15,22)(19,23)(21,24);(10,16,17,18)(11,21)(15,20,23,24)(19,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=egh,d^2=g,f^2=gh,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309481, #72309495, #72309508, … |
| maximal normal subgroup of | #72309481, #72309495, #72309508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36152);