#153244

order	512
nilpotency class	5
isoclinism	isoclinic class of #153086
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33607	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,3)(5,6)(7,12)(8,13)(9,28)(10,30)(11,17)(14,24)(15,29)(16,25)(18,34)(19,49)(20,36)(21,54)(22,37)(23,38)(26,45)(27,41)(31,63)(32,47)(33,52)(35,48)(39,59)(40,58)(42,57)(43,62)(44,60)(46,61)(50,66)(51,64)(53,68)(55,65)(56,67);(1,2)(3,4)(5,7)(6,12)(8,23)(9,18)(10,21)(11,22)(13,38)(14,42)(15,36)(16,35)(17,37)(19,27)(20,29)(24,57)(25,48)(26,51)(28,34)(30,54)(31,55)(32,39)(33,44)(40,67)(41,49)(43,66)(45,64)(46,68)(47,59)(50,62)(52,60)(53,61)(56,58)(63,65);(1,3,4,2)(6,13)(7,18)(8,24)(9,29)(10,25)(12,32)(14,16)(15,45)(17,41)(19,50)(20,22)(21,55)(23,56)(26,62)(27,60)(28,39)(30,40)(31,61)(33,65)(34,49)(35,66)(36,38)(37,64)(42,54)(43,63)(44,46)(47,52)(51,57)(58,59)(67,68)

Polycyclic presentation:

2^9|c^2=d,e^2=gh,f^2=i,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,e^b=ei,f^b=fi,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60440);