#391979

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 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)(3,5)(4,6)(7,8)(10,15,18,27)(16,32,19,30)(17,31,35,23)(33,34,36,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,19,22,16)(12,23,26,31)(13,27)(14,18)(15,29)(17,25,35,39)(20,34)(21,32,28,30)(24,33)(36,38)(37,40);(2,6)(5,8)(9,11)(10,16)(12,24)(13,28)(14,22)(15,30)(17,34)(18,19)(20,25)(21,29)(23,36)(26,38)(27,32)(31,33)(35,40)(37,39);(3,7)(5,8)(9,12,13,25,14,26,29,39)(10,17,15,31,18,35,27,23)(11,20,21,24,22,37,28,38)(16,33,32,34,19,36,30,40)

Polycyclic presentation:

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

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, 299175);