#153378

order	512
nilpotency class	5
isoclinism	isoclinic class of #153378
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13494
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

182 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320657, #72321372, #72321469
maximal normal subgroup of	#72320657, #72321372, #72321469

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60574);