#269571

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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

1062 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=deh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176767);