#146980

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 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)(5,6,10,16,12,7,14,20,13,18,11,17,9,22,28,21)(8,19,26,35,27,23,30,33,24,34,29,36,25,15,31,32);(2,4)(5,7,11,21,12,18,28,16,13,22,10,20,9,6,14,17)(8,23,26,33,27,34,30,36,24,15,29,32,25,19,31,35);(5,8,9,25,13,24,12,27)(6,15,7,19,18,23,22,34)(10,26,28,31,11,29,14,30)(16,32,20,35,17,33,21,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237207, #72238781, #72238789, …
maximal normal subgroup of	#72237207, #72238781, #72238789, …
maximal quotient of	#72237209

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54176);