#215455

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122651);