Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| 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×D4 | C24 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,22)(13,25)(14,17)(16,23)(18,28)(19,35)(20,33)(21,30)(24,34)(26,31)(27,32)(29,36)(37,40)(38,39);(2,5)(6,7)(9,11,14,21)(10,15,17,30)(12,19,24,37)(13,26,28,20)(16,29,32,39)(18,33,25,31)(22,35,34,40)(23,36,27,38);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,32)(18,34)(20,38)(21,37)(22,25)(23,28)(26,36)(30,39)(31,40)(33,35);(1,3,4,8)(2,6,5,7)(10,17)(15,30)(16,32)(18,25)(22,34)(29,39)(31,33)(35,40)Polycyclic presentation:
2^9|b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,f^c=fi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72317534, #72317559, #72317581, … |
| maximal normal subgroup of | #72317534, #72317559, #72317581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407702);