#416852

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

780 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:

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

Polycyclic presentation:

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

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, 324048);