#9015487

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	#44906	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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)(3,8)(4,7)(5,6)(9,10,15,13)(11,16,21,19)(12,17,23,14)(18,20,22,24)(25,26,31,34)(27,35,38,29)(28,33,30,32)(36,39,37,40);(3,4)(7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23)(25,27,31,38)(26,29,34,35)(28,36,30,37)(32,39,33,40);(1,3)(2,7)(4,6)(5,8)(9,12,15,23)(10,17,13,14)(11,18,21,22)(16,20,19,24)(25,27)(26,29)(28,36)(30,37)(31,38)(32,39)(33,40)(34,35);(3,4)(7,8)(9,13,15,10)(11,19,21,16)(12,17,23,14)(18,20,22,24)(25,28,31,30)(26,32,34,33)(27,36,38,37)(29,39,35,40);(1,4,6,3)(2,7,5,8)(9,14,15,17)(10,12,13,23)(11,20,21,24)(16,22,19,18)(25,29)(26,27)(28,39)(30,40)(31,35)(32,36)(33,37)(34,38);(1,5,6,2)(3,7,4,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23)(25,30,31,28)(26,33,34,32)(27,37,38,36)(29,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922683);