#8746950

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	#42602	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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)(3,8)(4,6)(5,7)(9,10,14,13)(11,15,18,16)(12,17,20,21)(19,22,24,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,6,5)(2,7,4,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,22)(20,24)(21,23)(26,27)(31,33)(32,34)(38,39);(1,3,6,5)(2,7,4,8)(11,18)(15,16)(19,24)(22,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(3,5)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,6,2)(3,8,5,7)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,16,18,15)(12,17,20,21)(19,22,24,23)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8654146);