#389325

order	512
nilpotency class	3
isoclinism	isoclinic class of #389324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14539
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296521);