#9006689

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

4475 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)(6,8)(9,10,15,18)(11,16,13,14)(12,17,20,24)(19,23,22,21)(25,26,30,32)(27,33,34,29)(28,31,36,39)(35,40,37,38);(3,7)(6,8)(9,11,15,13)(10,16,18,14)(12,19,20,22)(17,23,24,21)(25,26)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40);(1,2,5,4)(3,6,7,8)(11,13)(12,20)(14,16)(17,24)(27,34)(28,36)(29,33)(31,39);(9,12)(10,17)(11,19)(13,22)(14,21)(15,20)(16,23)(18,24)(25,27)(26,29)(28,37)(30,34)(31,40)(32,33)(35,36)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,14,18,16)(12,19,20,22)(17,23,24,21)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,4,5,2)(3,8,7,6)(9,14)(10,11)(12,21)(13,18)(15,16)(17,19)(20,23)(22,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,e^2=i,f^2=h,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, 8913885);