#265137

order	512
nilpotency class	3
isoclinism	isoclinic class of #265099
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|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,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172333);