#395761

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1771 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23,24,16)(13,17)(14,18)(19,26)(20,34,35,29)(21,30)(22,31)(25,37,38,32)(27,28)(33,39,40,36);(2,5)(6,8)(9,11,14,22)(10,15,18,31)(12,20,24,35)(13,26,28,21)(16,29,23,34)(17,19,27,30)(25,39,38,36)(32,33,37,40);(1,3,4,7)(2,6,5,8)(9,10)(11,19)(12,16)(13,27)(14,18)(15,26)(17,28)(20,33)(21,31)(22,30)(23,24)(25,37)(29,39)(32,38)(34,36)(35,40);(9,12)(10,16)(11,20)(13,25)(14,24)(15,29)(17,32)(18,23)(19,33)(21,36)(22,35)(26,39)(27,37)(28,38)(30,40)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302957);