#151973

order	512
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2177
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12637
prime radical	C₂×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=ghi,e^2=i,f^2=gi,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gh,h^a=hi,c^b=ch,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72223669, #72223679, C₂×#151973, …
maximal normal subgroup of	#72223669, #72223679, C₂×#151973, …
maximal quotient of	#72223668, #72223677, C₂×#151973, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59169);