#8760836

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	#41712	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4208 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=h,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, 8668032);