#8682299

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

4744 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,f^2=g,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, 8589495);