#263666

order	512
nilpotency class	3
isoclinism	isoclinic class of #263635
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×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

844 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:

(1,2,5,8)(3,9,11,16)(4,13,10,7)(6,14,15,12)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)(17,19,20,29)(18,23,25,27)(21,30,22,28)(24,32,26,31);(3,11)(6,15)(9,16)(12,14)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)(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=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170862);