#153556

order	512
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂×#2262	C₂
LCS	G ▷ 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

186 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 74 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)(13,31)(14,36)(15,38)(16,39)(18,23)(20,49)(21,53)(22,43)(25,59)(26,48)(27,32)(29,58)(30,33)(34,63)(35,64)(37,65)(40,41)(42,51)(44,55)(45,60)(46,47)(50,56)(52,68)(54,57)(61,62)(66,67);(5,6)(7,18)(8,23)(9,27)(10,30)(11,13)(12,17)(14,19)(15,24)(16,28)(20,50)(21,54)(22,44)(25,60)(26,55)(29,61)(31,43)(32,49)(33,53)(34,48)(35,58)(36,40)(37,59)(38,41)(39,42)(45,51)(46,57)(47,56)(52,65)(62,68)(63,66)(64,67);(6,13)(12,31)(14,34)(15,35)(16,30)(18,44)(23,55)(27,61)(32,62)(33,39)(36,63)(37,56)(38,64)(45,54)(50,65)(57,60)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60752);