#409298

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
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

918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316494);