#353918

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,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, 261114);