#454364

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1913 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)(3,4)(6,7)(9,10,15,20)(11,16,24,33)(12,25,27,36)(13,19,14,18)(17,29,34,26)(21,31,38,39)(22,30,37,40)(23,35,28,32);(5,6,8,7)(9,11,15,24)(10,16,20,33)(12,22,27,37)(13,28,14,23)(17,31,34,39)(18,35,19,32)(21,29,38,26)(25,30,36,40);(2,4)(6,7)(9,12)(10,17)(11,21)(13,29)(14,26)(15,27)(16,30)(18,36)(19,25)(20,34)(22,28)(23,37)(24,38)(31,35)(32,39)(33,40);(5,7,8,6)(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,31)(18,25)(19,36)(20,34)(21,28)(23,38)(24,37)(30,35)(32,40)(33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361560);