#130017

order	512
nilpotency class	4
isoclinism	isoclinic class of #129789
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37213);