#133767

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#541 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 614 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)(3,13,16,41,14,42,44,64)(4,10,20,36,18,37,49,62)(7,8,23,28,25,29,53,55)(9,30,32,48,33,56,59,19)(15,26,46,38,43,31,57,58)(17,35,22,52,45,40,47,63)(21,27,51,54,50,34,60,61);(2,8)(4,18)(9,26)(10,34)(12,29)(13,40)(15,43)(19,48)(20,49)(21,50)(27,37)(30,56)(31,33)(35,42)(36,62)(41,64)(46,57)(51,60)(52,63)(54,61);(1,3,7,17)(2,9,8,26)(4,19,21,15)(5,22,23,16)(6,14,25,45)(10,35,27,13)(11,38,28,32)(12,33,29,31)(18,48,50,43)(20,46,51,30)(24,47,53,44)(34,42,37,40)(36,41,54,52)(39,58,55,59)(49,57,60,56)(61,63,62,64)

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,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, 40963);