#8932867

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4568 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840063);