#8757420

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	C₂×#540	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)(11,15)(12,18)(13,16)(14,22)(17,20)(19,21)(23,24)(26,29)(27,33)(32,36)(35,39);(1,3,6,5)(2,7,4,8)(9,11)(10,17)(12,19)(13,15)(14,23)(16,20)(18,24)(21,22)(25,26,30,29)(27,34,33,31)(28,36,37,32)(35,38,39,40);(1,3,6,5)(2,8,4,7)(9,12,16,14)(10,18,13,22)(11,19,20,23)(15,21,17,24)(25,27,30,33)(26,31,29,34)(28,35,37,39)(32,38,36,40);(9,13,16,10)(11,15,20,17)(12,18,14,22)(19,24,23,21)(25,26,30,29)(27,31,33,34)(28,32,37,36)(35,38,39,40);(1,4,6,2)(3,7,5,8)(9,14,16,12)(10,18,13,22)(11,19,20,23)(15,24,17,21)(25,28,30,37)(26,32,29,36)(27,35,33,39)(31,38,34,40);(1,5,6,3)(2,7,4,8)(9,15)(10,11)(12,21)(13,20)(14,24)(16,17)(18,19)(22,23)(25,29,30,26)(27,34,33,31)(28,36,37,32)(35,40,39,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664616);