#286169

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24750	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 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)(3,6)(4,5)(7,8)(9,10,13,16,12,15,14,11)(17,18,20,24,21,25,26,23)(19,22,28,31,29,27,32,30);(2,5)(6,8)(9,11,13,10,12,16,14,15)(17,19,26,32,21,29,20,28)(18,22,23,30,25,27,24,31);(1,3,4,7)(2,6,5,8)(10,11)(13,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,29)(24,26)(28,31)(30,32);(2,5)(6,8)(17,20,21,26)(18,23,25,24)(19,28,29,32)(22,30,27,31)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=ehi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193365);