#152547

order	512
nilpotency class	5
isoclinism	isoclinic class of #151115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,g^b=gh,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59743);