#186785

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2020 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)(6,11)(7,14)(8,9)(13,17)(15,16)(18,19)(21,22,26,30,27,31,25,29)(23,32,33,24,28,35,34,36);(2,4)(5,6)(7,12)(8,11)(9,17)(10,13)(14,18)(15,19)(16,20)(21,23,27,28)(22,24,31,36)(25,33,26,34)(29,35,30,32);(5,7,10,16)(6,12,13,20)(8,14,9,15)(11,18,17,19)(21,24)(22,28)(23,31)(25,32)(26,35)(27,36)(29,33)(30,34);(5,8,10,9)(6,11,13,17)(7,15,16,14)(12,19,20,18)(21,25,27,26)(22,29,31,30)(23,33,28,34)(24,35,36,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93981);