#9063315

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	#35405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4354 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970511);