#146411

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=di,e^b=ei,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237201, #72237223, #72237256, …
maximal normal subgroup of	#72237201, #72237223, #72237256, …
maximal quotient of	#72237214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53607);