#444941

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

989 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:

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

Polycyclic presentation:

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

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, 352137);