#470683

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16693
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

830 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 377879);