#141500

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 317 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)(3,15,11,31)(4,20,25,51)(5,27,30,19)(6,32)(7,13)(8,14)(9,17)(10,24,38,50)(12,36)(16,43)(18,47,33,29)(21,53)(22,46)(23,55)(26,52,35,62)(28,58)(34,37)(39,59,45,63)(40,64)(41,61)(42,54,48,49)(44,60)(56,57);(1,3,15,2,9,19,30,13,8,11,31,14,7,27,5,17)(4,21,41,39,35,40,23,38,25,56,60,45,26,16,28,10)(6,33,42,37,36,54,47,46,22,18,48,12,34,49,29,32)(20,50,55,64,52,59,61,53,51,24,58,43,62,63,44,57);(1,4,22,50,9,35,34,59,8,25,6,24,7,26,36,63)(2,10,37,51,13,39,46,62,14,38,12,20,17,45,32,52)(3,16,33,61,19,21,54,58,11,40,18,44,27,56,49,55)(5,28,47,57,15,41,48,64,30,23,29,53,31,60,42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=ei,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, 48696);