#8172715

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	#38035	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4286 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8079911);