Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1732 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1732 conjugacy classes of subgroups — too many to draw. Only its 206 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 — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,17)(13,18,26,16)(14,19)(20,28)(21,29,36,27)(22,30)(23,32)(24,34)(25,33,38,31)(35,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,22)(10,15,19,30)(12,23,24,20)(13,21,26,36)(16,27,18,29)(17,32,34,28)(25,37,38,35)(31,40,33,39);(3,8)(5,7)(10,16)(12,24)(13,26)(15,27)(17,33)(18,19)(20,23)(21,36)(28,40)(29,30)(31,34)(32,39);(2,6)(5,7)(9,12,14,24)(10,17,19,34)(11,20,22,23)(13,25,26,38)(15,28,30,32)(16,31,18,33)(21,35,36,37)(27,39,29,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72341294, #72341409, #72341415, … |
| maximal normal subgroup of | #72341294, #72341409, #72341415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403400);