#8323392

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

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,f^2=gi,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, 8230588);