#133777

order	512
nilpotency class	4
isoclinism	isoclinic class of #133374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2815	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40973);