#141499

order	512
nilpotency class	4
isoclinism	isoclinic class of #141499
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9785
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#33607	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

325 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,15,11,30)(4,20,24,51)(5,26,29,19)(6,31)(7,13)(8,14)(9,17)(10,23,37,50)(12,36)(16,43)(18,47,32,28)(21,53)(22,54)(25,52,35,62)(27,57)(33,40)(34,46)(38,58,45,63)(39,64)(41,61)(42,59,48,49)(44,60)(55,56);(1,3,15,2,9,19,29,13,8,11,30,14,7,26,5,17)(4,21,41,38,35,39,22,37,24,55,60,45,25,16,27,10)(6,32,42,40,36,59,47,46,34,18,48,12,33,49,28,31)(20,50,54,64,52,58,61,53,51,23,57,43,62,63,44,56);(1,4,6,23,9,35,36,63,8,24,34,50,7,25,33,58)(2,10,12,20,13,38,31,52,14,37,40,51,17,45,46,62)(3,16,18,44,19,21,49,54,11,39,32,61,26,55,59,57)(5,27,28,53,15,41,42,43,29,22,47,56,30,60,48,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=gh,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48695);