#109029

order	512
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#390
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

622 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ci

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, 16225);