#351428

order	512
nilpotency class	3
isoclinism	isoclinic class of #348358
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

576 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)(3,5)(4,6)(7,8)(10,14)(11,18)(12,15)(16,20)(17,24)(21,23)(25,26,28,32)(27,34,35,30)(29,31,37,33)(36,39,40,38);(1,3,4,7)(2,5,6,8)(9,10,15,20,13,16,12,14)(11,17,22,24,18,23,19,21);(2,6)(5,8)(10,16)(14,20)(17,23)(21,24)(25,27,28,35)(26,30,32,34)(29,36,37,40)(31,38,33,39);(9,11,12,19,13,18,15,22)(10,17,14,21,16,23,20,24)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=efhi,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ch,g^d=ghi

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, 258624);