#475292

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	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=h,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 382488);