#8505283

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412479);