#352862

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#30258	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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)(3,6)(4,5)(7,8)(10,16)(11,12)(13,14)(17,18,22,24)(19,25,28,26)(20,21,29,30)(23,31,32,27);(2,5)(6,8)(9,10,13,16,15,12,14,11)(17,19,21,27,22,28,30,31)(18,23,20,25,24,32,29,26);(9,11,13,10,15,16,14,12)(18,24)(20,29)(23,32)(25,26);(1,3,4,7)(2,6,5,8)(9,12,13,11,15,10,14,16)(17,20,21,24,22,29,30,18)(19,26,27,23,28,25,31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260058);