#137214

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9021
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 437 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44410);