#9523579

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=i,f^2=g,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, 9430775);