#139727

order	512
nilpotency class	4
isoclinism	isoclinic class of #139724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,15,20,16,21,14,35)(11,22,26,50,27,51,37,49)(12,28,30,18,31,45,36,46)(13,32,23,47,34,19,38,40)(17,39,42,33,43,58,66,25)(24,53,56,70,57,71,55,48)(29,61,52,44,64,60,63,65)(41,62,68,72,69,54,59,67);(1,2)(3,5)(4,6)(7,8)(9,11,23,39)(10,17,40,50)(12,24,52,67)(13,33,15,37)(14,26,34,25)(16,27,38,58)(18,41,60,53)(19,22,20,42)(21,43,47,49)(28,59,65,70)(29,62,30,55)(31,57,63,72)(32,51,35,66)(36,56,64,54)(44,48,45,68)(46,69,61,71);(1,3)(2,5)(4,8)(6,7)(9,12,16,31)(10,18,21,46)(11,24,27,57)(13,29,34,64)(14,36,15,30)(17,41,43,69)(19,44,32,65)(20,45,35,28)(22,48,51,70)(23,52,38,63)(25,54,33,62)(26,56,37,55)(39,67,58,72)(40,60,47,61)(42,68,66,59)(49,71,50,53)

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,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,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, 46923);