#149512

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₁₆	C₂³
commutator, socle	C₂²	C₂×C₄×C₁₆
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,c^a=ci,c^b=cf

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56708);