#441147

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,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, 348343);