#153245

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	C₂×#41	D₈
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

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

normal subgroups only

This group has 263 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,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, 60441);