#9271817

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

4352 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9179013);