#8687332

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

4230 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,22)(23,24)(25,26,31,36)(27,32,30,35)(28,34,29,33)(37,40,38,39);(1,3,5,6)(2,4,7,8)(9,11,15,13)(10,16,20,17)(12,23,19,21)(14,22,18,24)(25,27,31,30)(26,32,36,35)(28,38,29,37)(33,40,34,39);(9,11,15,13)(10,17,20,16)(12,21,19,23)(14,22,18,24)(25,28,31,29)(26,33,36,34)(27,37,30,38)(32,39,35,40);(1,2)(3,4)(5,7)(6,8)(9,12,15,19)(10,18,20,14)(11,21,13,23)(16,22,17,24)(25,26)(27,35)(28,33)(29,34)(30,32)(31,36)(37,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,16,20,17)(12,23,19,21)(14,24,18,22)(25,29,31,28)(26,34,36,33)(27,37,30,38)(32,39,35,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,18)(10,19,20,12)(11,22,13,24)(16,21,17,23)(25,30,31,27)(26,35,36,32)(28,38,29,37)(33,40,34,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8594528);