#7196822

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
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	#3147	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

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

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, 7104018);