#263927

order	512
nilpotency class	3
isoclinism	isoclinic class of #263899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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)(3,12)(4,8)(5,13)(6,10)(7,11)(9,15)(14,16)(17,18)(19,24)(20,30)(21,26)(22,27)(23,28)(25,31)(29,32);(1,3,6,13,7,4,15,14)(2,5,9,12,11,16,10,8)(17,19)(18,24)(20,28)(21,29)(22,25)(23,30)(26,32)(27,31);(1,4)(2,8)(3,7)(5,9)(6,14)(10,16)(11,12)(13,15)(17,20,23,24)(18,25,28,29)(19,27,30,26)(21,31,22,32);(1,5,15,8,7,16,6,12)(2,3,10,14,11,4,9,13)(17,21,23,22)(18,26,28,27)(19,29,30,25)(20,31,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171123);