#153348

order	512
nilpotency class	5
isoclinism	isoclinic class of #153348
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13474
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	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

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

normal subgroups only

This group has 207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321384, #72321446
maximal normal subgroup of	#72320815, #72321384, #72321446

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60544);