#454341

order	512
nilpotency class	3
isoclinism	isoclinic class of #454029
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

1279 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,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,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, 361537);