#134564

order	512
nilpotency class	4
isoclinism	isoclinic class of #124987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1536
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#8579
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,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, 41760);