#8936472

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	#43914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4298 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8843668);