#9607343

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
prime radical	#46381	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9514539);