#133266

order	512
nilpotency class	4
isoclinism	isoclinic class of #133265
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8522
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#3449	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

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

normal subgroups only

This group has 586 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,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,f^c=fh,f^d=fi,f^e=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40462);