#9435169

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=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, 9342365);