#8271271

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	#44912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178467);