#316412

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
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

653 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)(9,10,14,19)(11,21,12,16)(13,20,15,18)(17,22,24,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,14,12)(10,16,19,21)(13,23,15,22)(17,20,24,18)(25,27,34,37,28,35,29,36)(26,30,38,40,31,33,32,39);(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18)(25,27,28,35)(26,30,31,33)(29,37,34,36)(32,40,38,39);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)(26,31)(30,33)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 223608);