#146410

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

896 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237201, #72237227, #72237260, …
maximal normal subgroup of	#72237201, #72237227, #72237260, …
maximal quotient of	#72237214, #72307616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53606);