#6464096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371292);