#356099

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2350 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=de,e^d=ei,h^d=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263295);