#153242

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	#33610	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

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

normal subgroups only

This group has 269 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)(7,19)(8,14)(9,13)(10,16)(11,31)(12,21)(15,28)(17,29)(18,26)(20,34)(22,53)(23,44)(24,58)(25,37)(27,45)(30,38)(32,55)(33,65)(35,57)(36,51)(39,62)(40,61)(41,59)(42,60)(43,63)(46,49)(47,56)(48,50)(52,64)(54,66)(67,68);(1,2)(3,4)(5,7)(6,12)(8,21)(9,22)(10,20)(11,24)(13,36)(14,19)(15,33)(16,44)(17,35)(18,34)(23,26)(25,51)(27,52)(28,56)(29,50)(30,49)(31,64)(32,38)(37,53)(39,67)(40,46)(41,47)(42,48)(43,66)(45,58)(54,62)(55,61)(57,60)(59,65)(63,68);(1,3,4,2)(6,13)(7,20)(9,28)(10,26)(11,29)(12,32)(14,37)(15,17)(16,40)(18,38)(19,46)(21,23)(22,54)(24,52)(25,59)(27,60)(30,63)(31,39)(33,58)(34,53)(35,67)(36,44)(41,42)(43,45)(47,64)(48,66)(49,56)(51,68)(55,65)(61,62)

Polycyclic presentation:

2^9|c^2=di,e^2=gh,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,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, 60438);