#395061

order	512
nilpotency class	3
isoclinism	isoclinic class of #394022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 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)(3,6)(4,5)(7,8)(11,21)(13,14)(16,29)(18,19)(22,31)(23,37)(24,25)(26,28)(30,36)(32,34)(35,38)(39,40);(2,5)(6,8)(9,10,15,20)(11,22,25,29)(12,17,27,33)(13,19,14,18)(16,21,31,24)(23,36,38,39)(26,34,28,32)(30,35,40,37);(9,11)(10,16)(12,23)(13,24)(14,21)(15,25)(17,30)(18,22)(19,29)(20,31)(26,35)(27,38)(28,37)(32,39)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,23)(13,28)(14,26)(15,27)(16,30)(18,34)(19,32)(20,33)(21,35)(22,36)(24,37)(25,38)(29,39)(31,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313113, #72313116, #72313122
maximal normal subgroup of	#72313113, #72313116, #72313122

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302257);