#415755

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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)(3,8)(4,6)(5,7)(10,15,18,26)(11,19,22,27)(17,33,34,30)(21,37,38,35);(1,3,5,4)(2,6,7,8)(9,10,14,18)(11,20,22,29)(12,17,25,34)(13,26,28,15)(16,27,32,19)(21,36,38,39)(23,30,24,33)(31,37,40,35);(1,4,5,3)(2,6,7,8)(9,11,14,22)(10,16,18,32)(12,21,25,38)(13,27,28,19)(15,29,26,20)(17,31,34,40)(23,35,24,37)(30,39,33,36);(9,12,13,24,14,25,28,23)(10,17,26,33,18,34,15,30)(11,21,27,37,22,38,19,35)(16,31,20,36,32,40,29,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 322951);