#353548

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1818 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,23)(17,28)(22,36)(24,32)(25,31)(27,34)(29,33)(30,39)(35,38)(37,40);(5,6,7,8)(9,11,14,24,15,25,13,23)(10,16,18,31,20,32,19,21)(12,22,29,39,28,38,27,37)(17,30,33,36,26,40,34,35);(9,12,13,27,15,28,14,29)(10,17,19,34,20,26,18,33)(11,22,23,37,25,38,24,39)(16,30,21,35,32,40,31,36);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,31,32,21)(17,33,26,34)(22,37,38,39)(30,36,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260744);