#496981

order	512
nilpotency class	4
isoclinism	isoclinic class of #483928
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	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

559 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)(11,21)(12,26,28,61)(13,18)(14,33)(15,37)(16,23)(17,29,43,64)(19,36)(20,32)(22,51,52,71)(24,56)(25,41)(27,63,54,59)(30,47,38,62)(31,60)(34,68)(35,48,67,44)(39,70,49,53)(40,69)(42,57,72,65)(45,66)(46,58)(50,55);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,27,28,54)(13,23)(14,34)(15,25)(17,42,43,72)(18,21)(19,46)(20,41)(22,38,52,30)(24,31)(26,59,61,63)(29,65,64,57)(32,55)(33,56)(35,53,67,70)(36,69)(37,50)(39,48,49,44)(40,45)(47,71,62,51)(58,66)(60,68);(2,5)(6,8)(9,12)(10,17)(11,22)(13,28)(14,35)(15,38)(16,39)(18,43)(19,47)(20,48)(21,49)(23,52)(24,57)(25,27)(26,60)(29,66)(30,32)(31,67)(33,61)(34,65)(36,64)(37,44)(40,63)(41,42)(45,62)(46,59)(50,72)(51,68)(53,69)(54,55)(56,71)(58,70);(1,3,4,7)(2,6,5,8)(10,18)(11,23)(17,43)(19,45)(20,37)(22,52)(24,34)(25,55)(26,61)(27,54)(33,60)(44,48)(47,62)(53,70)(57,65)(58,69)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=ei,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ceh,d^b=di,f^b=fi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404177);