#444438

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,h^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^c=hi,e^d=ei,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, 351634);