#414512

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

798 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,26)(13,29)(14,20)(15,21)(16,24)(17,27)(18,33)(19,30)(23,32)(25,36)(28,38)(31,40)(34,37)(35,39);(1,3,5,4)(2,6,7,8)(9,11,15,25)(10,16,21,37)(12,18,28,29)(13,17,33,39)(14,34,30,24)(19,36,20,22)(23,35,40,27)(26,31,38,32);(1,4,5,3)(2,6,7,8)(9,12,15,28)(10,17,21,39)(11,23,25,40)(13,22,33,36)(14,35,30,27)(16,32,37,31)(18,24,29,34)(19,38,20,26);(9,13,30,31,15,33,14,32)(10,18,19,23,21,29,20,40)(11,17,24,38,25,39,34,26)(12,22,27,16,28,36,35,37)

Polycyclic presentation:

2^9|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,e^a=ei,f^a=fi,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, 321708);