#293982

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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)(4,5)(6,8)(9,10,14,16)(11,12,15,13)(17,18,23,27)(19,24,25,31)(20,30,32,29)(21,26,22,28);(2,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,23,25)(18,24,27,31)(20,28,32,26)(21,29,22,30);(9,11,12,16,14,15,13,10)(17,20,22,24,23,32,21,31)(18,25,26,30,27,19,28,29);(1,3,4,8)(2,6,5,7)(17,21,23,22)(18,26,27,28)(19,29,25,30)(20,31,32,24)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=f,d^2=ehi,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, 201178);