#134568

order	512
nilpotency class	4
isoclinism	isoclinic class of #124865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1525
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8583
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

293 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,f^b=fi,e^c=eh,f^c=fghi,g^c=gi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41764);