#353952

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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 ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1986 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)(11,20)(12,24)(13,27)(14,19)(15,18)(16,22)(17,25)(21,30)(23,32)(26,34)(28,31)(29,33)(35,36)(37,40)(38,39);(2,4)(9,11,14,22,15,23,13,28)(10,16,19,32,18,31,27,20)(12,21,25,37,26,38,29,36)(17,30,34,40,33,39,24,35);(5,7,8,6)(9,12)(10,17)(11,21)(13,29)(14,25)(15,26)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,27)(28,36)(31,39)(32,40);(10,18)(16,31)(17,33)(19,27)(20,32)(24,34)(30,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,e^b=ei,f^b=fi,e^c=ei,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, 261148);