Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,14)(11,16,20,30)(12,22,25,35)(13,26,29,18)(17,28,31,24)(19,36,37,40)(21,34,27,33)(23,38,32,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,27)(14,30)(15,20)(17,19)(18,34)(21,29)(22,38)(24,40)(25,32)(26,33)(28,36)(31,37)(35,39);(9,12)(10,17)(11,19)(13,28)(14,31)(15,25)(16,32)(18,35)(20,37)(21,39)(22,26)(23,30)(24,29)(27,38)(33,40)(34,36);(2,6)(5,7)(11,20)(16,30)(19,37)(21,27)(23,32)(33,34)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,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 | #72212257, #72212272, #72267418, … |
| maximal normal subgroup of | #72212257, #72212272, #72267418, … |
| maximal quotient of | #72212048, #72212212, #72266852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407241);