#353961

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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1930 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)(5,6,8,7)(9,10,13,19,15,20,14,18)(11,21,25,31,23,32,24,16)(12,26,28,34,29,33,27,17)(22,30,37,35,36,39,38,40);(2,4)(9,11,14,24,15,23,13,25)(10,16,18,32,20,31,19,21)(12,22,27,38,29,36,28,37)(17,30,33,40,34,39,26,35);(5,7,8,6)(9,12,13,28,15,29,14,27)(10,17,19,26,20,34,18,33)(11,22,25,37,23,36,24,38)(16,30,21,35,31,39,32,40);(11,23)(16,31)(21,32)(22,36)(24,25)(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,h^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	#72320944, #72320952, #72321009
maximal normal subgroup of	#72320944, #72320952, #72321009
maximal quotient of	#72321552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261157);