#153329

order	512
nilpotency class	5
isoclinism	isoclinic class of #153329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2254
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13452
prime radical	C₂³	#509
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

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

normal subgroups only

This group has 209 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)(3,14,16,44)(4,20)(5,11)(6,27)(7,31)(8,13)(9,17,33,49)(10,25)(12,29)(15,46,19,42)(18,53,50,64)(21,54)(22,45,56,40)(23,41)(24,43)(26,61)(28,36,48,63)(30,37)(32,47,39,51)(34,58,62,59)(35,57)(38,60)(52,55);(3,15,16,19)(4,21)(9,32,33,39)(10,35)(14,42,44,46)(17,51,49,47)(18,48,50,28)(20,54)(22,58,56,59)(24,26)(25,57)(34,40,62,45)(36,64,63,53)(38,55)(43,61)(52,60);(1,3,7,15,8,16,5,19)(2,9,11,39,13,33,31,32)(4,22,26,59,21,56,24,58)(6,28,29,18,30,48,23,50)(10,36,38,64,35,63,55,53)(12,40,27,34,41,45,37,62)(14,43,42,54,44,61,46,20)(17,52,47,57,49,60,51,25)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224986, #72224999, #72225003, …
maximal normal subgroup of	#72224986, #72224999, #72225003, …
maximal quotient of	C₂×#153329

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60525);