#267657

order	512
nilpotency class	3
isoclinism	isoclinic class of #267563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

851 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=gi,f^d=fh

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, 174853);