#9445234

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

4523 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gh,f^2=gh,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, 9352430);