#10045654

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9952850);