#150874

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 263 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,14,21,17,25,46,24)(11,26,29,66,32,68,45,52)(12,33,36,63,39,59,13,40)(15,47,35,22,16,49,42,23)(18,51,53,28,57,27,56,69)(19,43,20,37,61,44,60,38)(30,58,48,67,31,72,50,62)(34,55,41,64,70,54,71,65);(1,3)(2,5)(4,8)(6,7)(9,11,17,32)(10,18,25,57)(12,34,39,70)(13,41,36,71)(14,45,46,29)(15,48,16,50)(19,58,61,72)(20,62,60,67)(21,56,24,53)(22,64,23,65)(26,40,68,63)(27,38,51,37)(28,43,69,44)(30,42,31,35)(33,66,59,52)(47,54,49,55);(3,8)(5,7)(9,12,35,44,14,36,16,38,17,39,42,43,46,13,15,37)(10,19,47,63,24,60,23,33,25,61,49,40,21,20,22,59)(11,27,48,41,29,69,31,70,32,51,50,71,45,28,30,34)(18,52,64,62,56,68,55,72,57,66,65,67,53,26,54,58)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ghi,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,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, 58070);