#8749660

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,e^2=h,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, 8656856);