#316364

order	512
nilpotency class	3
isoclinism	isoclinic class of #262731
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=h,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223560);