#145881

order	512
nilpotency class	4
isoclinism	isoclinic class of #145881
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10578
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#454
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 210 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=dg,c^2=dh,d^2=i,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=eg,f^c=fh,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, 53077);