#470740

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

753 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)(3,5)(4,6)(7,8)(9,10,14,19,13,20,15,18)(11,16,28,31,22,32,23,36)(12,24,26,17,27,37,29,35)(21,38,25,34,40,33,39,30);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,25)(13,22)(14,23)(15,28)(17,33)(18,31)(19,36)(20,32)(21,26)(24,30)(27,39)(29,40)(34,37)(35,38);(9,12,14,26,13,27,15,29)(10,17,18,24,20,35,19,37)(11,21,23,39,22,40,28,25)(16,30,31,38,32,34,36,33);(2,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,34)(18,36)(19,31)(20,32)(24,38)(25,26)(27,40)(29,39)(30,35)(33,37)

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377936);