#8699216

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

4409 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)(11,17)(12,13)(14,18)(15,16)(19,22)(20,21)(23,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,7)(4,8)(5,6)(10,16)(12,17)(18,24)(20,22)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,13)(10,17,16,12)(14,19,23,21)(18,22,24,20)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(2,6)(4,7)(9,12)(10,11)(13,16)(14,22)(15,17)(18,21)(19,24)(20,23)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3)(2,7)(4,6)(5,8)(9,13,15,11)(10,12,16,17)(14,19,23,21)(18,22,24,20)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,4,5,7)(2,8,6,3)(9,14,15,23)(10,18,16,24)(11,19,13,21)(12,20,17,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8606412);