#153360

order	512
nilpotency class	5
isoclinism	isoclinic class of #153215
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2240
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13480
prime radical	C₂×#508	C₂²
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

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

normal subgroups only

This group has 200 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:

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

Polycyclic presentation:

2^9|a^2=e,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, #72321401, #72321417
maximal normal subgroup of	#72321265, #72321401, #72321417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60556);