#489808

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	#61290	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18,23,28)(19,29,32,59)(20,25,34,49)(21,37,42,57)(22,43,40,27)(24,45,47,31)(26,52,55,41)(30,60,61,58)(33,65,44,53)(35,67,39,72)(36,70,66,51)(38,56,74,48)(46,64,76,62)(50,75,54,69)(63,80,71,78)(68,79,73,77);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,15,10,11)(17,19,23,32)(18,24,28,47)(20,30,34,61)(21,38,42,74)(22,44,40,33)(25,46,49,76)(26,53,55,65)(27,56,43,48)(29,57,59,37)(31,52,45,41)(35,68,39,73)(36,71,66,63)(50,78,54,80)(51,79,70,77)(58,67,60,72)(62,75,64,69);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,20)(18,25)(19,30)(21,39)(22,36)(23,34)(24,46)(26,54)(27,51)(28,49)(29,58)(31,64)(32,61)(33,63)(35,42)(37,72)(38,73)(40,66)(41,75)(43,70)(44,71)(45,62)(47,76)(48,77)(50,55)(52,69)(53,80)(56,79)(57,67)(59,60)(65,78)(68,74);(2,9)(7,16)(8,10)(11,15)(20,34)(25,49)(30,61)(35,39)(36,66)(46,76)(50,54)(51,70)(58,60)(62,64)(63,71)(67,72)(68,73)(69,75)(77,79)(78,80)

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fhi,g^a=gi,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, 397004);