#8509974

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

4709 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^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, 8417170);