#8688206

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8595402);