#263846

order	512
nilpotency class	3
isoclinism	isoclinic class of #263819
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=ei,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,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171042);