#417073

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,f^c=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, 324269);