#440973

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1073 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:

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

Polycyclic presentation:

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

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, 348169);