#469335

order	512
nilpotency class	3
isoclinism	isoclinic class of #454970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1213 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)(13,26)(14,29)(15,20)(17,25)(18,28)(19,30)(21,35)(22,33)(24,31)(27,34)(32,36)(37,38)(39,40);(2,5)(6,8)(9,11,15,12)(10,16,20,23)(13,21,28,24)(14,25,30,22)(17,19,33,29)(18,31,26,35)(27,37,40,36)(32,34,38,39);(1,3)(2,6)(4,7)(5,8)(9,12,15,11)(10,17,20,33)(13,24,28,21)(14,22,30,25)(16,29,23,19)(18,32,26,38)(27,36,40,37)(31,39,35,34);(3,7)(6,8)(9,13)(10,18,20,26)(11,21)(12,24)(14,27)(15,28)(16,31,23,35)(17,32,33,38)(19,34,29,39)(22,36)(25,37)(30,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,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, 376531);