#8720344

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

4300 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627540);