#409345

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

825 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316541);