#186746

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
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×#3511	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2360 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)(7,15)(8,11)(9,13)(10,14)(12,16)(17,20)(18,19)(21,22,25,30,26,29,24,28)(23,31,32,27,34,35,33,36);(2,4)(6,11)(8,9)(12,19)(13,14)(15,20)(16,17)(21,23,26,34)(22,27,29,36)(24,32,25,33)(28,35,30,31);(5,7,10,18)(6,12,14,20)(8,16,9,17)(11,19,13,15)(21,24,26,25)(22,28,29,30)(23,32,34,33)(27,35,36,31);(5,8,10,9)(6,11,14,13)(7,16,18,17)(12,19,20,15)(22,29)(27,36)(28,30)(31,35)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93942);