#416908

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 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)(3,7)(4,5)(6,8)(9,10,15,28,14,19,13,18)(11,16,21,30,22,31,26,33)(12,23,27,17,24,34,25,32)(20,35,37,40,36,39,38,29);(2,5)(6,7)(9,11)(10,16)(12,20,24,36)(13,26)(14,22)(15,21)(17,29,32,40)(18,33)(19,31)(23,39,34,35)(25,38,27,37)(28,30);(9,12)(10,17)(11,20)(13,27)(14,24)(15,25)(16,29)(18,34)(19,32)(21,38)(22,36)(23,28)(26,37)(30,39)(31,40)(33,35);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,g^b=gi,e^c=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, 324104);