#9062127

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

4581 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)(9,10,15,12)(11,16,14,19)(13,17,21,20)(18,23,22,24)(25,26,31,34)(27,32,35,30)(28,29,33,40)(36,37,38,39);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,12,19)(13,18,21,22)(17,23,20,24)(25,27,31,35)(26,32,34,30)(28,38,33,36)(29,39,40,37);(1,3,5,8)(2,6,4,7)(10,12)(11,14)(13,21)(23,24)(27,35)(28,33)(29,40)(30,32);(1,4,5,2)(3,7,8,6)(9,12,15,10)(11,16,14,19)(13,17,21,20)(18,24,22,23)(25,28)(26,29)(27,36)(30,39)(31,33)(32,37)(34,40)(35,38);(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)(25,29)(26,33)(27,37)(28,34)(30,36)(31,40)(32,38)(35,39);(9,14,15,11)(10,16,12,19)(13,22,21,18)(17,23,20,24)(25,30)(26,27)(28,39)(29,36)(31,32)(33,37)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,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, 8969323);