#134869

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	#8623
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,d^2=gi,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^c=fhi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287633, #72319310, #72319322
maximal normal subgroup of	#72287633, #72319310, #72319322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42065);