#8932669

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	#46454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^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, 8839865);