#111187

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

666 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

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, 18383);