#371875

order	512
nilpotency class	3
isoclinism	isoclinic class of #371873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14082
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

1075 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,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, 279071);