#153602

order	512
nilpotency class	6
isoclinism	isoclinic class of #153601
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator, Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	#13694	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

181 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60798);