#131326

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8403
prime radical	#60749	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

737 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38522);