#8186972

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

4457 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)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2,4,5)(3,6,7,8)(10,11)(15,17)(16,18)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(3,7)(6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094168);