#146409

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
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	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×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) = 64

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

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

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,g^a=gi,e^b=ei,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237207, #72237229, #72237259, …
maximal normal subgroup of	#72237207, #72237229, #72237259, …
maximal quotient of	#72237218, #72239346, #72280424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53605);