#111413

order	512
nilpotency class	3
isoclinism	isoclinic class of #102079
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 18609);