#151022

order	512
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#106
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1367
prime radical	C₂³	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,d^2=f,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ci,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58218);