#109167

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#380
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

592 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,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, 16363);