#357963

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265159);