#111027

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

716 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239599
maximal normal subgroup of	#72239599
maximal quotient of	#72239608

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18223);