#8201520

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

4327 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8108716);