#8348264

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

4405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8255460);