#8550963

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

4413 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)(10,11)(15,17)(16,18)(22,23)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(3,7)(6,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27,31,34)(26,32,30,29)(28,35,37,40)(33,39,38,36);(1,2,5,4)(3,6,7,8)(12,19)(15,17)(20,24)(22,23)(27,34)(29,32)(35,40)(36,39);(1,3,5,7)(2,6,4,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,4,5,2)(3,8,7,6)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458159);