#502836

order	512
nilpotency class	4
isoclinism	isoclinic class of #501172
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	#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

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,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, 410032);