Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| 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 | #24414 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
970 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)(4,6)(5,8)(9,10,13,15)(11,19,20,33)(12,17,23,27)(14,18,25,28)(16,29,26,22)(21,32,34,40)(24,31,36,38)(30,39,37,35);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(16,26)(17,27)(18,28)(21,34)(30,37)(31,38);(9,11,14,22)(10,16,18,19)(12,21,24,35)(13,20,25,29)(15,26,28,33)(17,30,31,32)(23,34,36,39)(27,37,38,40);(2,6)(5,7)(9,12,13,23)(10,17,15,27)(11,21,20,34)(14,24,25,36)(16,30,26,37)(18,31,28,38)(19,32,33,40)(22,35,29,39)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72317526, #72317552, #72317559, … |
| maximal normal subgroup of | #72317526, #72317552, #72317559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409920);