#472565

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 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:

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

Polycyclic presentation:

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

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, 379761);