#293887

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=e,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,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 201083);