#133277

order	512
nilpotency class	4
isoclinism	isoclinic class of #133277
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8524
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#3512	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40473);