#137196

order	512
nilpotency class	4
isoclinism	isoclinic class of #12170
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9019
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

519 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gi,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281031, #72281057, #72281172
maximal normal subgroup of	#72281031, #72281057, #72281172

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44392);