#473971

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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)(9,10,15,20,14,19,13,17)(11,21,25,34,22,16,24,30)(12,26,29,33,28,32,27,18)(23,37,39,35,36,40,38,31);(1,3,4,7)(2,5,6,8)(9,11,15,24,14,22,13,25)(10,16,20,34,19,21,17,30)(12,23,27,39,28,36,29,38)(18,31,32,37,33,35,26,40);(3,7)(5,8)(10,17,19,20)(11,22)(16,30,21,34)(18,32,33,26)(23,36)(24,25)(31,37,35,40)(38,39);(9,12)(10,18,19,33)(11,23)(13,29)(14,28)(15,27)(16,31,21,35)(17,26,20,32)(22,36)(24,39)(25,38)(30,40,34,37)

Polycyclic presentation:

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

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, 381167);