#134865

order	512
nilpotency class	4
isoclinism	isoclinic class of #134747
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#405	C₂³
socle	C₂	#8621
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=g,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fi,f^c=fhi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287633, #72343284, #72343285
maximal normal subgroup of	#72287633, #72343284, #72343285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42061);