#291118

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59229	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

943 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)(3,7)(4,6)(5,8)(17,18)(19,24)(20,30)(21,26)(22,28)(23,27)(25,31)(29,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(14,16)(17,19,23,25)(18,24,27,31)(20,26,32,28)(21,29,22,30);(9,10)(11,16)(12,14)(13,15)(17,20,22,31,23,32,21,24)(18,25,26,30,27,19,28,29);(2,6)(5,7)(10,14)(13,16)(17,21,23,22)(18,26,27,28)(19,29,25,30)(20,24,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198314);