#153376

order	512
nilpotency class	5
isoclinism	isoclinic class of #153376
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

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

normal subgroups only

This group has 181 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321372, #72321450
maximal normal subgroup of	#72320815, #72321372, #72321450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60572);