#409297

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,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, 316493);