#146026

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10616
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

220 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=dg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53222);