#8811220

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8718416);