#7182824

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#537	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7090020);