#153414

order	512
nilpotency class	5
isoclinism	isoclinic class of #153215
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2240
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13537
prime radical	C₂³	#510
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 157 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,c^2=d,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fg,g^b=gh,h^b=hi,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321265, #72321372, #72321401
maximal normal subgroup of	#72321265, #72321372, #72321401

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60610);