#473850

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

917 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)(3,6)(4,5)(7,8)(9,10,15,19,13,18,14,28)(11,20,23,32,24,16,21,30)(12,25,26,33,27,17,29,38)(22,31,35,40,36,34,37,39);(2,5)(6,8)(9,11,14,23,13,24,15,21)(10,16,28,30,18,20,19,32)(12,22,26,37,27,36,29,35)(17,31,38,39,25,34,33,40);(1,3)(2,6)(4,7)(5,8)(11,21)(14,15)(16,30)(17,25)(19,28)(20,32)(22,35)(23,24)(26,29)(31,40)(34,39)(36,37);(3,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,26)(15,29)(16,31)(18,25)(19,33)(20,34)(21,35)(23,37)(24,36)(28,38)(30,39)(32,40)

Polycyclic presentation:

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

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, 381046);