#8339903

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	#37587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=hi,e^2=h,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, 8247099);