Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3496 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,17)(11,15,20,30)(12,22,24,16)(14,18,26,34)(19,29,37,40)(21,31,27,35)(23,39,38,32)(25,33,28,36);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,23)(13,20)(14,27)(16,32)(17,30)(18,35)(19,25)(21,26)(22,39)(24,38)(28,37)(29,33)(31,34)(36,40);(1,3,5,4)(2,7,8,6)(9,12,13,24)(10,16,17,22)(11,19)(14,28)(15,29)(18,36)(20,37)(21,38,27,23)(25,26)(30,40)(31,39,35,32)(33,34);(1,4,5,3)(2,7,8,6)(11,20)(15,30)(19,37)(21,27)(23,38)(29,40)(31,35)(32,39)Polycyclic presentation:
2^9|a^2=ei,b^2=e,c^2=ef,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72311241, #72311268, #72311272, … |
| maximal normal subgroup of | #72311241, #72311268, #72311272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410785);