#415265

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

997 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322461);