#139753

order	512
nilpotency class	4
isoclinism	isoclinic class of #139751
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

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, 46949);