#152826

order	512
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2233
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13131
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=dh,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225337, #72225570, #72225573
maximal normal subgroup of	#72225337, #72225570, #72225573
maximal quotient of	#72225531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60022);