#9028331

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

4485 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935527);