#9440949

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

4369 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)(25,26,31,32)(27,28,29,33)(30,38,39,34)(35,40,37,36);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(26,32)(27,29)(34,38)(35,37);(1,3)(2,7)(4,6)(5,8)(10,15)(13,23)(16,24)(17,20)(25,27,31,29)(26,33,32,28)(30,35,39,37)(34,40,38,36);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,28,31,33)(26,27,32,29)(30,36,39,40)(34,35,38,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=g,f^2=h,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, 9348145);