#9203689

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

4464 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110885);