#269169

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176365);