#9041019

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

4415 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8948215);