#137817

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9086
prime radical	C₂⁴	#131
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=di,e^c=ei,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45013);