#202754

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ei,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109950);