#139741

order	512
nilpotency class	4
isoclinism	isoclinic class of #139741
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	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23898	C₂
LCS	G ▷ C₂²×D₄ ▷ 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)(3,13,18,40)(4,19)(5,23,14,10)(6,26)(7,12)(8,16)(9,21)(11,29)(15,42)(17,39,50,61)(20,52,41,62)(22,36)(24,32)(25,35,28,37)(27,34)(30,57,59,64)(31,47)(33,53,55,54)(38,48)(43,60,46,51)(44,58)(45,63)(49,56);(1,3,14,16,7,18,5,24)(2,8,23,40,12,32,10,13)(4,15,41,46,22,45,20,43)(6,17,25,49,29,50,28,47)(9,30,55,58,34,59,33,38)(11,31,35,39,26,56,37,61)(19,51,62,42,36,60,52,63)(21,44,53,64,27,48,54,57);(1,4)(2,9)(3,15)(5,20)(6,27)(7,22)(8,30)(10,33)(11,36)(12,34)(13,38)(14,41)(16,46)(17,48)(18,45)(19,26)(21,29)(23,55)(24,43)(25,54)(28,53)(31,60)(32,59)(35,52)(37,62)(39,63)(40,58)(42,61)(44,50)(47,64)(49,57)(51,56)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46937);