#406106

order	512
nilpotency class	3
isoclinism	isoclinic class of #406106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14924
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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)(3,12)(4,16,18,23)(5,11)(6,10)(7,14)(8,17,24,30)(9,19)(13,20,27,31)(15,21)(22,28)(25,26,29,32);(1,3,6,15)(2,7)(4,13,18,27)(5,14,19,28)(8,20)(9,12)(10,22)(11,21)(16,29)(17,26,30,32)(23,25)(24,31);(1,2)(3,7)(4,8)(5,9)(6,10)(11,19)(12,14)(13,20)(15,22)(16,30)(17,23)(18,24)(21,28)(25,26)(27,31)(29,32);(1,4,5,17,6,18,19,30)(2,8,9,23,10,24,11,16)(3,13,14,26,15,27,28,32)(7,20,12,25,22,31,21,29)

Polycyclic presentation:

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

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, 313302);