#469168

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16648
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

821 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)(3,13)(4,9,18,27)(5,19,21,10)(6,22)(7,12)(8,16)(11,20)(14,24,29,32)(15,26,31,23)(17,28)(25,30);(1,3,6,16,7,17,20,30)(2,8,11,13,12,25,22,28)(4,14,21,31,18,29,5,15)(9,26,19,32,27,23,10,24);(1,4)(2,9,12,27)(3,14)(5,20)(6,21)(7,18)(8,23,25,26)(10,11,19,22)(13,24,28,32)(15,30)(16,31)(17,29);(1,5,6,4,7,21,20,18)(2,10,22,27,12,19,11,9)(3,15,16,14,17,31,30,29)(8,24,28,23,25,32,13,26)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376364);