#153351

order	512
nilpotency class	5
isoclinism	isoclinic class of #153351
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13476
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=de,b^2=dgh,d^2=ghi,e^2=g,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dg,e^c=egh,f^c=fgh,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321384, #72321469
maximal normal subgroup of	#72320815, #72321384, #72321469

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60547);