#153601

order	512
nilpotency class	6
isoclinism	isoclinic class of #153601
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator, Frattini, cosocle	C₂×C₃₂	C₂³
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

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

normal subgroups only

This group has 201 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) = 128

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

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

Polycyclic presentation:

2^9|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.

Relationships

maximal subgroup of	#72286862
maximal normal subgroup of	#72286862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60797);