#8701286

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

4423 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=gh,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, 8608482);