#146983

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
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×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

558 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)(5,6,10,17,12,16,30,35,13,19,11,18,9,15,29,34)(7,20,22,8,21,26,36,24,23,33,31,27,28,14,32,25);(2,4)(5,7,11,22,12,21,29,36,13,23,10,31,9,28,30,32)(6,14,18,25,16,20,34,8,19,26,17,24,15,33,35,27);(5,8,9,25,13,27,12,24)(6,7,16,21,19,23,15,28)(10,26,29,20,11,14,30,33)(17,22,35,36,18,31,34,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237201, #72239904, #72239910, …
maximal normal subgroup of	#72237201, #72239904, #72239910, …
maximal quotient of	#72237205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54179);