#94746

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#106
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	#7455
prime radical	C₂²	#1422
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

55 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,13,7,15,30,48,8,16,31,49,22,51,56,64,9,14,32,47,35,11,37,38,27,40,54,62,23,41,28,58,10,17,33,50,36,39,59,61,26,12,43,44,21,45,55,63,6,34,29,60,19,42,53,46,4,24,25,57,3,18,20,52);(1,3,4,19,6,21,26,36,10,23,27,35,9,22,8,7)(2,11,12,18,14,39,24,51,17,42,16,41,34,15,40,45)(5,28,25,37,29,56,43,30,33,20,54,53,32,55,31,59)(13,46,44,58,47,48,57,63,50,38,49,52,60,61,62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225395, #72225425, #72225473, …
maximal normal subgroup of	#72225395, #72225425, #72225473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1942);