#137456

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
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	#9045
prime radical	#3054	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

370 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fhi,g^c=gi,h^c=hi,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, 44652);