#8406657

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

4738 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)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,30,31)(27,34,28,32)(29,33,37,38)(35,40,36,39);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(26,31)(32,34)(33,38)(39,40);(1,2)(3,6)(4,5)(7,8)(12,20)(15,22)(18,24)(21,23)(25,27,30,28)(26,32,31,34)(29,36,37,35)(33,40,38,39);(1,3,4,8)(2,6,5,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28,30,27)(26,32,31,34)(29,36,37,35)(33,39,38,40);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^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, 8313853);