#316613

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
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

614 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)(9,10,12,15)(11,17,19,24)(13,22,21,16)(14,23,18,20)(25,26,31,35)(27,32,28,38)(29,39,30,34)(33,37,40,36);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,16)(14,19)(17,23)(20,24)(21,22)(25,27,29,37,31,28,30,36)(26,32,39,40,35,38,34,33);(9,11,12,19)(10,14,15,18)(13,23,21,20)(16,24,22,17)(25,28,31,27)(26,33,35,40)(29,37,30,36)(32,39,38,34);(2,6)(5,7)(10,15)(14,18)(16,22)(17,24)(25,29,31,30)(26,34,35,39)(27,36,28,37)(32,40,38,33)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,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,d^b=dh,f^c=fh,g^c=gi

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, 223809);