#8407020

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

4623 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314216);