#8943980

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

4481 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:

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

Polycyclic presentation:

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

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, 8851176);