#9057327

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

4341 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:

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

Polycyclic presentation:

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

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, 8964523);