#475693

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17064
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 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)(3,9)(4,16,19,23)(5,8)(6,11)(7,12)(10,20,28,18)(13,29,30,27)(14,24)(15,26)(17,31,32,25)(21,22);(2,8,12,11)(9,24,26,22)(10,23,28,16)(25,29,31,27);(1,3,6,14,7,15,5,21)(2,9,11,24,12,26,8,22)(4,17,20,30,19,32,18,13)(10,27,16,25,28,29,23,31);(1,4,5,18,7,19,6,20)(2,10,11,16,12,28,8,23)(3,13,14,17,15,30,21,32)(9,25,22,27,26,31,24,29)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=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, 382889);