#9007858

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	#43417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,e^2=ghi,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, 8915054);