#268416

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,20,9,14,10,13)(7,8,16,19,12,18,15,17)(21,22,26,31)(23,32,24,28)(25,29,35,30)(27,34,36,33);(1,2)(3,4)(5,7,9,12)(6,8,14,18)(10,16,11,15)(13,19,20,17)(21,23,26,24)(22,27,31,36)(25,34,35,33)(28,29,32,30);(2,4)(5,8)(6,12)(7,14)(9,18)(10,19)(11,17)(13,15)(16,20)(21,24,26,23)(22,28,31,32)(25,34,35,33)(27,29,36,30);(10,11)(13,20)(15,16)(17,19)(22,29)(25,35)(27,32)(28,36)(30,31)(33,34)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=i,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

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, 175612);