#133771

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	C₂×#538	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 627 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,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, 40967);