#129305

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fg,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36501);