Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
707 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 707 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,16)(13,25,26,17)(14,18)(15,21,28,37)(20,34)(22,33)(23,31,38,29)(24,30)(27,35,39,40)(32,36);(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,19)(18,28)(23,35)(24,36)(25,33)(26,37)(29,34)(30,39)(31,32)(38,40);(9,12,14,24)(10,16,18,30)(11,20,22,36)(13,23,26,38)(15,27,28,39)(17,31,25,29)(19,32,33,34)(21,35,37,40)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=egi,f^b=fg,d^c=dg,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281844, #72281857, #72281877, … |
| maximal normal subgroup of | #72281844, #72281857, #72281877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36796);