#132963

order	512
nilpotency class	4
isoclinism	isoclinic class of #132529
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 493 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:

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

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=eghi,f^b=fgh,d^c=dgh,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, 40159);