#415756

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

659 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)(9,10)(11,21,25,16)(12,23,29,17)(13,18)(14,20)(15,19)(22,31)(24,37,32,27)(26,30)(28,38,33,35)(34,36)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,26)(13,23)(14,32)(15,25)(17,30)(18,29)(19,21)(20,37)(22,28)(24,34)(27,36)(31,35)(33,40)(38,39);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,33)(15,29)(16,31)(18,24)(19,23)(20,38)(21,39)(25,40)(26,32)(28,34)(30,37)(35,36);(9,13,14,31,15,30,34,39)(10,18,20,22,19,26,36,40)(11,23,32,35,25,17,24,38)(12,27,33,16,29,37,28,21)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,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, 322952);