#9015493

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

4270 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)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(3,4)(7,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23)(26,30)(28,36)(31,33)(34,40);(1,3)(2,7)(4,6)(5,8)(9,12,15,23)(10,17,13,14)(11,18,21,22)(16,20,19,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(3,4)(7,8)(9,13,15,10)(11,19,21,16)(12,17,23,14)(18,20,22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,6,3)(2,7,5,8)(9,14,15,17)(10,12,13,23)(11,20,21,24)(16,22,19,18)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(1,5,6,2)(3,7,4,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=hi,e^2=i,f^2=gh,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, 8922689);