#142700

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	#33765	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 346 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:

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

Polycyclic presentation:

2^9|b^2=h,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, 49896);