#473989

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 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)(3,6)(4,5)(7,8)(9,10,15,19,14,18,21,23)(11,20,17,34,24,13,29,30)(12,16,27,32,28,33,36,37)(22,26,25,31,38,35,39,40);(2,5)(6,8)(9,11,21,29,14,24,15,17)(10,13,23,34,18,20,19,30)(12,25,27,38,28,39,36,22)(16,31,32,35,33,40,37,26);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,22,24,38)(13,26,20,35)(14,28)(15,27)(17,25,29,39)(18,33)(19,32)(21,36)(23,37)(30,31,34,40);(3,7)(6,8)(9,13)(10,17)(11,23)(12,26)(14,20)(15,30)(16,25)(18,29)(19,24)(21,34)(22,37)(27,31)(28,35)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,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, 381185);