#376207

order	512
nilpotency class	3
isoclinism	isoclinic class of #376205
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14175
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eh,e^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,g^b=gi,e^c=ei,h^c=hi,f^d=fi

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, 283403);