#263814

order	512
nilpotency class	3
isoclinism	isoclinic class of #263799
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	#18272	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,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,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,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, 171010);