Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 576 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,20,15,18,13,19)(11,21,23,35,24,16,29,32)(12,25,27,34,28,36,30,17)(22,37,39,33,26,40,38,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,26)(13,29)(14,23)(15,24)(17,33)(18,21)(19,32)(20,35)(22,28)(25,37)(27,38)(30,39)(31,34)(36,40);(9,12)(10,17,18,34)(11,22)(13,27)(14,30)(15,28)(16,31,21,33)(19,25,20,36)(23,38)(24,26)(29,39)(32,40,35,37)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37362);