Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×2−1+4 | C23 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
825 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 825 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2,5,8)(3,7,4,6)(9,10)(11,20)(12,17)(13,25)(14,28)(15,19)(16,22)(18,29)(21,37)(23,32)(24,36)(26,33)(27,34)(30,40)(31,38)(35,39);(1,3,5,4)(2,6,8,7)(11,21)(14,29)(16,31)(18,28)(20,37)(22,38)(23,30)(24,27)(26,39)(32,40)(33,35)(34,36);(9,11)(10,16)(12,22)(13,26)(14,30)(15,23)(17,32)(18,35)(19,33)(20,25)(21,29)(24,39)(27,38)(28,31)(34,37)(36,40);(1,4,5,3)(2,7,8,6)(9,12,15,13)(10,17,19,25)(11,22,23,26)(14,24,29,27)(16,32,33,20)(18,34,28,36)(21,38,30,39)(31,40,35,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72271508 |
| maximal normal subgroup of | #72271508 |
| maximal quotient of | #72271518 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408167);