#129306

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,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,e^c=ei,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, 36502);