#8709186

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

4535 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,f^2=g,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, 8616382);