#315747

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 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)(10,11)(12,16)(13,14)(17,18)(19,25)(20,29)(21,23)(22,27)(24,26)(28,31)(30,32);(1,3)(2,5)(4,7)(6,8)(9,10,13,11,15,12,14,16)(17,19,26,32,21,28,20,27)(18,22,29,31,23,30,24,25);(3,7)(5,8)(9,11,13,12,15,16,14,10)(18,23)(22,30)(24,29)(25,31);(9,12,13,16,15,10,14,11)(17,19,26,32,21,28,20,27)(18,22,29,31,23,30,24,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=ehi,c^2=fh,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222943);