#9015501

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

4624 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=ghi,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, 8922697);