#9481452

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

3516 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^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, 9388648);