#8891277

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4105 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)(11,14)(12,20)(13,17)(15,19)(16,22)(18,24)(21,23)(25,26,31,36)(27,37,39,32)(28,35,30,33)(29,34,38,40);(1,3,6,5)(2,7,4,8)(9,11,13,19)(10,14,17,15)(12,21,22,18)(16,24,20,23)(25,26)(27,37)(28,33)(29,40)(30,35)(31,36)(32,39)(34,38);(1,3,6,5)(2,8,4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,27,31,39)(26,32,36,37)(28,38,30,29)(33,40,35,34);(9,10)(11,14)(12,16)(13,17)(15,19)(18,23)(20,22)(21,24)(25,28,31,30)(26,33,36,35)(27,38,39,29)(32,40,37,34);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,23)(21,22)(25,29)(26,34)(27,28)(30,39)(31,38)(32,33)(35,37)(36,40);(1,5,6,3)(2,7,4,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,20)(19,21)(25,30,31,28)(26,35,36,33)(27,29,39,38)(32,34,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798473);