#249817

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 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:

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

Polycyclic presentation:

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

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, 157013);