#8509519

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

4506 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8416715);