#441339

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=gi,c^2=efi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348535);