#133027

order	512
nilpotency class	4
isoclinism	isoclinic class of #132573
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

476 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40223);