#441145

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1441 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348341);