#206122

order	512
nilpotency class	3
isoclinism	isoclinic class of #19621
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1126 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)(7,14)(8,13)(9,18)(10,12)(11,15)(16,20)(17,19)(21,22)(23,29)(24,26)(25,33)(27,32)(28,30)(31,35)(34,36);(2,4)(5,7,9,15,10,16,8,17)(6,11,13,14,12,19,18,20)(22,26)(27,34)(28,33)(29,35);(6,12)(11,19)(13,18)(14,20)(21,23,30,36,24,31,25,32)(22,27,33,35,26,34,28,29);(5,8,10,9)(6,13,12,18)(7,15,16,17)(11,20,19,14)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113318);