#8709159

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

4647 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)(3,7)(4,5)(6,8)(25,26,29,32)(27,33,35,30)(28,31,38,37)(34,40,36,39);(2,5)(6,7)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27,29,35)(26,30,32,33)(28,34,38,36)(31,39,37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,23)(18,24)(25,27,29,35)(26,30,32,33)(28,34,38,36)(31,39,37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,27)(26,30)(28,36)(29,35)(31,40)(32,33)(34,38)(37,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,26)(27,30)(28,37)(29,32)(31,38)(33,35)(34,40)(36,39);(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,37)(33,40)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^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, 8616355);