#126438

order	512
nilpotency class	4
isoclinism	isoclinic class of #126437
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7800
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fgi,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33634);