#460619

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

985 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)(3,5)(4,6)(7,8)(9,10,13,18,14,19,26,16)(11,20,22,15,23,28,38,30)(12,17,24,31,25,33,27,32)(21,35,36,29,37,34,39,40);(1,3,4,7)(2,5,6,8)(9,11,14,23)(10,15)(12,21,25,37)(13,22,26,38)(16,20)(17,29)(18,28)(19,30)(24,36,27,39)(31,34)(32,35)(33,40);(2,6)(5,8)(10,16)(12,24)(13,26)(15,28)(18,19)(20,30)(21,36)(22,38)(25,27)(29,40)(31,32)(37,39);(9,12,14,25)(10,17,19,33)(11,21,23,37)(13,27,26,24)(15,29,30,40)(16,31,18,32)(20,34,28,35)(22,39,38,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,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, 367815);