#475133

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=e,d^2=g,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, 382329);