#256780

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46232	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,32)(30,31);(1,3,5,4)(2,6,7,8)(9,10,12,16)(11,15,13,14)(20,29)(22,31)(23,28)(27,30);(1,4,5,3)(2,6,7,8)(10,14)(11,13)(15,16)(17,19,21,26)(18,22,24,31)(20,30,29,27)(23,32,28,25);(2,7)(6,8)(9,11,12,13)(10,15,16,14)(19,26)(22,31)(25,32)(27,30)

Polycyclic presentation:

2^9|b^2=e,c^2=eh,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163976);