#265017

order	512
nilpotency class	3
isoclinism	isoclinic class of #264939
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	C₂×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172213);