#454335

order	512
nilpotency class	3
isoclinism	isoclinic class of #454021
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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361531);