#8110490

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

4478 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)(6,8)(13,21)(16,20)(18,24)(22,23)(25,26,32,31)(27,35,30,33)(28,38,34,29)(36,37,39,40);(3,7)(6,8)(10,12)(15,17)(16,20)(22,23)(25,27,32,30)(26,33,31,35)(28,39,34,36)(29,40,38,37);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(25,28)(26,29)(27,36)(30,39)(31,38)(32,34)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,29,32,38)(26,34,31,28)(27,37,30,40)(33,39,35,36);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,30,32,27)(26,35,31,33)(28,39,34,36)(29,40,38,37);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,31,32,26)(27,35,30,33)(28,38,34,29)(36,40,39,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017686);