#153330

order	512
nilpotency class	5
isoclinism	isoclinic class of #153330
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13452
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,13,15,42)(4,19)(5,25)(6,30)(7,10)(8,16,14,45)(9,22)(11,28)(12,27)(17,51,47,64)(18,52,31,44)(20,43,56,39)(21,38)(23,62)(24,53)(26,34,49,55)(29,35)(32,58,46,63)(33,57,40,50)(36,61)(37,59)(41,60)(48,54);(1,2)(3,14,15,8)(4,9)(5,11)(6,12)(7,10)(13,16,42,45)(17,46,47,32)(18,40,31,33)(19,22)(20,55,56,34)(21,35)(23,37)(24,36)(25,28)(26,43,49,39)(27,30)(29,38)(41,48)(44,50,52,57)(51,58,64,63)(53,61)(54,60)(59,62);(1,3,6,31,7,15,27,18)(2,8,12,40,10,14,30,33)(4,20,23,63,24,56,60,58)(5,26,28,17,29,49,21,47)(9,34,37,64,36,55,54,51)(11,39,25,32,38,43,35,46)(13,41,52,53,42,62,44,19)(16,48,57,61,45,59,50,22)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224988, #72224999, #72225001, …
maximal normal subgroup of	#72224988, #72224999, #72225001, …
maximal quotient of	C₂×#153330

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60526);