Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #44936 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#550 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
1003 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1003 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,19,21,24)(12,17,13,16)(14,18)(15,20)(22,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,19)(18,23);(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72292869, C2×#130482 |
| maximal normal subgroup of | #72292869, C2×#130482 |
| maximal quotient of | C2×#130482, #72292954, #72292955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37678);