#353957

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1970 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)(3,4)(9,10,14,18,15,19,13,27)(11,20,23,16,22,32,28,31)(12,24,25,17,26,33,29,34)(21,30,38,40,36,39,37,35);(2,4)(5,6,7,8)(9,11)(10,16)(12,21)(13,28)(14,23)(15,22)(17,30)(18,32)(19,31)(20,27)(24,35)(25,38)(26,36)(29,37)(33,40)(34,39);(9,12,14,25,15,26,13,29)(10,17,18,33,19,34,27,24)(11,21,23,38,22,36,28,37)(16,30,32,40,31,39,20,35);(11,22)(16,31)(20,32)(21,36)(23,28)(30,39)(35,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320952, #72320976, #72321006
maximal normal subgroup of	#72320952, #72320976, #72321006
maximal quotient of	#72321653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261153);