#348659

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3531	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

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, 255855);