#363022

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270218);