#146982

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
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

572 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)(5,6,11,18,13,17,31,35,9,16,12,19,10,28,30,24)(7,14,21,33,20,27,36,32,22,25,15,34,29,8,23,26);(2,4)(5,7,12,15,13,20,30,23,9,22,11,21,10,29,31,36)(6,14,18,33,17,27,35,32,16,25,19,34,28,8,24,26);(5,8,10,25,9,27,13,14)(6,15,17,23,16,21,28,36)(7,19,20,24,22,18,29,35)(11,26,30,34,12,32,31,33)

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,g^a=gi,c^b=ch,d^b=di,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	#72237201, #72239902, #72239911, …
maximal normal subgroup of	#72237201, #72239902, #72239911, …
maximal quotient of	#72237205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54178);