Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1523 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,23)(13,27)(14,16)(15,21)(17,26)(18,28)(20,37)(22,33)(24,32)(25,34)(29,40)(30,36)(31,39)(35,38);(1,3,4,8)(2,5,6,7)(11,20)(13,28)(15,31)(18,27)(19,37)(21,39)(22,29)(24,38)(25,30)(32,35)(33,40)(34,36);(9,11)(10,15)(12,24)(13,29)(14,22)(16,32)(17,33)(18,35)(19,23)(20,28)(21,26)(25,39)(27,31)(30,38)(34,37)(36,40);(10,16)(15,32)(17,23)(18,27)(19,33)(31,35)(34,36)(37,40)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,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=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267418, #72267431, #72267531, … |
| maximal normal subgroup of | #72267418, #72267431, #72267531, … |
| maximal quotient of | #72267422, #72267447, C2×#498990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406186);