#391254

order	512
nilpotency class	3
isoclinism	isoclinic class of #391252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14622
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

910 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298450);