#152370

order	512
nilpotency class	5
isoclinism	isoclinic class of #150138
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 ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,6)(4,5)(7,8)(9,10,13,20,16,22,15,38)(11,17,24,41,27,43,26,61)(12,28,31,50,34,18,33,44)(14,35,30,47,37,48,36,21)(19,46,45,29,49,63,62,32)(23,51,54,70,57,39,56,64)(25,58,53,67,60,68,59,42)(40,66,65,52,69,72,71,55);(2,5)(6,8)(9,11,16,27)(10,17,22,43)(12,23,34,57)(13,24,15,26)(14,25,37,60)(18,39,28,51)(19,40,49,69)(20,41,38,61)(21,42,47,67)(29,52,32,55)(30,53,36,59)(31,54,33,56)(35,58,48,68)(44,64,50,70)(45,65,62,71)(46,66,63,72);(1,3,4,7)(2,6,5,8)(10,18)(12,29)(13,15)(14,36)(17,39)(19,47)(20,50)(21,49)(22,28)(23,52)(24,26)(25,59)(30,37)(31,46)(32,34)(33,63)(35,45)(38,44)(40,67)(41,70)(42,69)(43,51)(48,62)(53,60)(54,66)(55,57)(56,72)(58,65)(61,64)(68,71)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59566);