#502807

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
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×Q₈	C₂⁴
prime radical	#25742	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

914 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)(11,19)(12,23)(13,18)(14,16)(15,30)(17,26)(20,28)(21,34)(22,32)(24,39)(25,31)(27,38)(29,36)(33,37)(35,40);(2,5)(6,7)(9,11,13,14)(10,16,18,19)(12,24,25,40)(15,29,28,22)(17,33,34,38)(20,36,30,32)(21,37,26,27)(23,35,31,39);(9,12,14,26,13,25,11,21)(10,17,19,23,18,34,16,31)(15,27,22,24,28,37,29,40)(20,35,32,33,30,39,36,38);(1,3,4,8)(2,6,5,7)(10,18)(16,19)(17,34)(20,30)(23,31)(32,36)(33,38)(35,39)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,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, 410003);