#263922

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	#18569	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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, 171118);