#152283

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2226
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12811
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

269 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 269 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225337, #72225340, #72225341, …
maximal normal subgroup of	#72225337, #72225340, #72225341, …
maximal quotient of	#72225461

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59479);