#475168

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

834 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)(11,16)(12,23,26,17)(13,15)(14,19)(18,28)(20,36,38,32)(21,31)(22,33)(24,34,40,39)(25,35,37,30)(27,29);(2,5)(3,7)(10,15,19,18)(16,31,33,29)(17,30,23,35)(32,39,36,34);(1,3)(2,6)(4,7)(5,8)(9,11,14,22)(10,16,19,33)(12,24,26,40)(13,27,28,21)(15,29,18,31)(17,34,23,39)(20,37,38,25)(30,36,35,32);(3,7)(6,8)(9,12,13,25,14,26,28,37)(10,17,15,30,19,23,18,35)(11,20,21,24,22,38,27,40)(16,32,31,39,33,36,29,34)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 382364);