#8700502

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	#36442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4407 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607698);