#153271

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2240
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13407
prime radical	#33716	C₂
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

303 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,18)(4,23)(5,16)(6,32)(7,36)(8,22)(9,13)(10,17)(11,20)(12,26)(14,35)(15,34)(19,53)(21,31)(24,52)(25,55)(27,47)(28,61)(29,59)(30,56)(33,54)(37,39)(38,50)(40,58)(41,57)(42,60)(43,49)(44,48)(45,63)(46,64)(51,62);(1,3)(2,11)(4,24)(5,15)(6,14)(7,37)(8,13)(9,20)(10,22)(12,38)(16,18)(17,34)(19,27)(21,39)(23,42)(25,61)(26,46)(28,59)(29,58)(30,49)(31,48)(32,44)(33,47)(35,56)(36,43)(40,57)(41,60)(45,53)(50,63)(51,64)(52,55)(54,62);(1,4,5,25,10,29,9,41)(2,12,13,45,17,47,16,51)(3,19,20,54,22,46,15,50)(6,33,30,53,7,38,31,64)(8,40,11,42,18,52,34,28)(14,24,48,60,37,58,49,61)(21,55,36,23,56,57,32,59)(26,44,62,35,27,43,63,39)

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fgi,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=egi,f^b=fg,g^b=gh,h^b=hi,e^c=egi,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	#72224986, #72225007, #72225011, …
maximal normal subgroup of	#72224986, #72225007, #72225011, …
maximal quotient of	C₂×#153271

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60467);