#502985

order	512
nilpotency class	4
isoclinism	isoclinic class of #501164
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,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,g^c=gh,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410181);