#490000

order	512
nilpotency class	4
isoclinism	isoclinic class of #489745
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61491	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fhi,g^a=gi,d^b=di,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397196);