#186719

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2236 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93915);