#8289428

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

4511 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8196624);