#8399367

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
prime radical	#45720	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4340 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8306563);