#139735

order	512
nilpotency class	4
isoclinism	isoclinic class of #139733
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	#3485	C₂²
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302901, #72302902, #72302916
maximal normal subgroup of	#72302901, #72302902, #72302916

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46931);