#8807368

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714564);