#416443

order	512
nilpotency class	3
isoclinism	isoclinic class of #396613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ 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)(9,10,14,18)(11,19,22,32)(12,23,25,16)(13,26,28,17)(15,27,30,21)(20,34,36,33)(24,31,40,38)(29,35,37,39);(9,11)(10,15)(12,20,25,36)(13,27)(14,22)(16,29,23,37)(17,32)(18,30)(19,26)(21,28)(24,39,40,35)(31,34,38,33);(2,5)(6,7)(9,12,13,24,14,25,28,40)(10,16,17,31,18,23,26,38)(11,20,21,35,22,36,27,39)(15,29,19,33,30,37,32,34);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323639);