Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
931 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 931 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,14,17,27,32)(11,19,22,33,21,15,28,30)(12,23,25,34,24,16,26,31)(20,35,37,40,36,29,38,39);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,29)(17,19)(18,33)(23,35)(24,36)(25,37)(26,38)(27,28)(30,32)(31,40)(34,39);(9,12,14,24)(10,16,17,23)(11,20,21,36)(13,26,27,25)(15,29,19,35)(18,34,32,31)(22,38,28,37)(30,40,33,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266604 |
| maximal normal subgroup of | #72266604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36956);