#415226

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,37)(29,32)(30,33)(34,38)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(25,27)(26,31)(28,35)(29,34)(30,36)(32,38)(33,40)(37,39);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(27,34)(28,29)(31,38)(32,37)(35,36)(39,40);(17,19,20,24)(18,21,22,23)(25,28,30,29)(26,32,33,37)(27,35,36,34)(31,38,40,39)

Polycyclic presentation:

2^9|a^2=ehi,b^2=ef,c^2=e,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322422);