#148485

order	512
nilpotency class	4
isoclinism	isoclinic class of #148483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11174
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,e^c=eh,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55681);