#93885

order	512
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#367
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂	#7101
prime radical	C₂²	#1373
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

57 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ceg,c^2=egi,d^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225362, #72225369, #72225473, …
maximal normal subgroup of	#72225362, #72225369, #72225473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1081);