#142701

order	512
nilpotency class	4
isoclinism	isoclinic class of #142700
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9950
commutator, Frattini, cosocle	C₂×#131	C₂³
prime radical	#3058	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=eg,f^b=fi,g^b=gi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,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, 49897);