#8107590

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	#42518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4426 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:

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

Polycyclic presentation:

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

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, 8014786);