#454918

order	512
nilpotency class	3
isoclinism	isoclinic class of #454110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 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)(9,10,13,19,15,21,14,20)(11,16,23,32,24,18,30,37)(12,25,27,36,29,35,28,17)(22,33,26,34,38,40,39,31);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,33)(18,21)(19,32)(20,37)(22,28)(25,34)(27,38)(29,39)(31,35)(36,40);(3,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,27)(15,29)(16,31)(18,34)(19,35)(20,25)(21,36)(23,39)(24,38)(26,30)(32,40)(33,37);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,22)(13,30)(14,23)(15,24)(16,21)(17,34)(19,32)(20,37)(25,33)(26,28)(27,39)(29,38)(31,36)(35,40)

Polycyclic presentation:

2^9|a^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,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, 362114);