Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130050 |
| 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 | C23×D4 | 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 | |
707 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 707 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5,6,7)(9,10)(11,19)(12,23,25,16)(13,26,27,17)(14,18)(15,21,29,36)(20,32,35,39)(22,33)(24,30)(28,34)(31,38)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,29)(12,20,25,35)(13,21,27,36)(16,28,23,37)(17,19,26,33)(24,34,38,40)(30,39,31,32);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,23)(19,32)(21,34)(22,35)(26,30)(27,38)(29,37)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227716, #72284102, #72285252, … |
| maximal normal subgroup of | #72227716, #72284102, #72285252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37247);