#371879

order	512
nilpotency class	3
isoclinism	isoclinic class of #371877
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 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)(4,13,17,9,16,23,18,24)(5,10,21,14,19,26,20,25)(15,28,29,22,30,27,31,32);(1,3,7,11,8,12,6,2)(4,9,18,23,16,24,17,13)(5,14,20,26,19,25,21,10)(15,22,29,28,30,32,31,27);(1,4,6,18,8,16,7,17)(2,9,3,13,11,24,12,23)(5,15,21,29,19,30,20,31)(10,22,14,27,26,32,25,28);(1,5,6,21,8,19,7,20)(2,10,12,25,11,26,3,14)(4,15,18,29,16,30,17,31)(9,22,23,28,24,32,13,27)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ef,d^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 279075);