#209186

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

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, 116382);