#122061

order	512
nilpotency class	3
isoclinism	isoclinic class of #97594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,g^b=ghi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29257);