#502840

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
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	#25741	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

932 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)(3,7)(4,5)(6,8)(11,19)(12,23)(13,25)(15,28)(17,31)(18,26)(20,34)(21,35)(22,36)(29,39)(30,40)(37,38);(2,5)(6,7)(9,10,14,16)(11,20,22,37)(12,17,23,31)(13,26,25,18)(15,29,30,21)(19,34,36,38)(24,33,27,32)(28,39,40,35);(9,11,12,19,14,22,23,36)(10,15,17,28,16,30,31,40)(13,21,27,39,25,29,24,35)(18,20,33,38,26,37,32,34);(1,3,4,8)(2,6,5,7)(10,16)(15,30)(17,31)(18,26)(20,37)(28,40)(32,33)(34,38)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=e,f^2=i,g^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^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, 410036);