#147380

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 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)(5,6,10,16,11,15,26,31,12,8,14,24,9,23,25,35)(7,17,20,33,21,22,28,34,19,32,30,36,18,13,27,29);(2,4)(5,7)(6,13,8,22)(9,18)(10,20)(11,21)(12,19)(14,30)(15,17,23,32)(16,29,24,34)(25,27)(26,28)(31,33,35,36);(5,8,10,24,11,23,26,35,12,6,14,16,9,15,25,31)(7,17,30,36,21,22,27,29,19,32,20,33,18,13,28,34)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,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	#72239200, #72239201, #72239249, …
maximal normal subgroup of	#72239200, #72239201, #72239249, …
maximal quotient of	#72239334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54576);