#480034

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1082 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,20,31)(12,16)(13,25,26,17)(14,18)(15,22)(21,32,33,39)(23,36,37,29)(24,30)(27,34)(28,35)(38,40);(9,11)(10,15,18,27)(12,21)(13,22)(14,20)(16,28,30,38)(17,19,25,31)(23,35)(24,33)(26,34)(29,32,36,39)(37,40);(11,20)(15,27)(19,31)(21,33)(22,34)(28,38)(32,39)(35,40);(9,12,14,24)(10,16,18,30)(11,21,20,33)(13,23,26,37)(15,28,27,38)(17,29,25,36)(19,32,31,39)(22,35,34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fgi,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387230);