#348488

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255684);