#8418211

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325407);