#366003

order	512
nilpotency class	3
isoclinism	isoclinic class of #366002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13984
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273199);