#9481525

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388721);