#8269847

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,32)(29,38)(30,34)(31,35)(33,39)(37,40);(1,2,4,6)(3,5,7,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(27,28)(29,39)(32,36)(33,38);(1,3)(2,5)(4,7)(6,8)(12,20)(15,22)(18,24)(21,23)(25,27,31,28)(26,32,35,36)(29,37,39,34)(30,33,40,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28,31,27)(26,32,35,36)(29,34,39,37)(30,33,40,38);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29,31,39)(26,33,35,38)(27,37,28,34)(30,32,40,36);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30,31,40)(26,34,35,37)(27,38,28,33)(29,32,39,36)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8177043);