#133055

order	512
nilpotency class	4
isoclinism	isoclinic class of #132549
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

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

normal subgroups only

This group has 475 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:

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

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=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fg,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, 40251);