#153252

order	512
nilpotency class	5
isoclinism	isoclinic class of #153081
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13381
prime radical	#33600	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 276 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)(3,13)(4,18)(5,24)(6,28)(7,10)(8,15)(9,21)(11,20)(12,26)(14,39)(16,48)(17,49)(19,53)(22,40)(23,50)(25,41)(27,60)(29,42)(30,43)(31,59)(32,62)(33,61)(34,58)(35,45)(36,56)(37,46)(38,47)(44,64)(51,55)(52,57)(54,63);(8,30)(13,39)(31,63)(32,38)(33,41)(37,53)(42,49)(48,64);(1,3,6,29,7,14,26,17)(2,8,12,38,10,30,28,32)(4,19,22,59,23,46,57,54)(5,25,20,16,27,61,56,44)(9,33,35,48,34,41,51,64)(11,37,24,31,36,53,60,63)(13,40,49,18,39,52,42,50)(15,45,62,21,43,55,47,58)

Polycyclic presentation:

2^9|c^2=dh,e^2=gh,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,d^c=di,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224986, #72224997, #72225001, …
maximal normal subgroup of	#72224986, #72224997, #72225001, …
maximal quotient of	C₂×#153252

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60448);