#268393

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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 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)(7,15,19,12,17,18,16,8)(21,22)(23,31)(24,33)(25,35)(26,28)(27,32)(29,34)(30,36);(1,2)(3,4)(10,11)(13,20)(15,18)(16,19)(21,23,26,24)(22,27,28,29)(25,34,36,32)(30,33,35,31);(2,4)(5,7)(6,12)(8,14)(9,17)(10,19)(11,16)(13,18)(15,20)(22,28)(27,29)(30,35)(31,33);(5,8,9,12)(6,7,14,17)(10,15,11,18)(13,19,20,16)(21,24,26,23)(22,29,28,27)(25,32,36,34)(30,31,35,33)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,d^2=h,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, 175589);