#497056

order	512
nilpotency class	4
isoclinism	isoclinic class of #483993
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=f,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404252);