#419039

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326235);