#8700245

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4287 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=g,e^2=h,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, 8607441);