#8987075

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

4443 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)(9,10,15,13)(11,12,19,16)(14,21,22,17)(18,23,24,20)(25,26,30,34)(27,28,32,31)(29,37,39,33)(35,40,38,36);(1,2,6,5)(3,8,7,4)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,3)(2,4)(5,8)(6,7)(10,13)(12,16)(14,22)(18,24)(25,27,30,32)(26,31,34,28)(29,38,39,35)(33,36,37,40);(1,2,6,5)(3,4,7,8)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(1,4,6,8)(2,7,5,3)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,28)(26,32)(27,34)(29,36)(30,31)(33,38)(35,37)(39,40);(1,5,6,2)(3,8,7,4)(9,14,15,22)(10,17,13,21)(11,18,19,24)(12,20,16,23)(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=g,c^2=gh,d^2=ghi,e^2=g,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, 8894271);