#9007871

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	#35470	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=g,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, 8915067);