#8505037

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

4542 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)(3,7)(9,10,16,14)(11,20,15,17)(12,19,13,18)(21,23,22,24)(26,32)(28,36)(29,37)(33,40);(1,2,4,5)(3,6,7,8)(9,11,16,15)(10,17,14,20)(12,21,13,22)(18,23,19,24)(25,26)(27,33)(28,34)(29,38)(30,36)(31,32)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,14,19)(11,21,15,22)(17,23,20,24)(25,27,31,39)(26,33,32,40)(28,37,36,29)(30,38,34,35);(1,2)(3,6)(4,5)(7,8)(9,13,16,12)(10,19,14,18)(11,21,15,22)(17,23,20,24)(25,28)(26,34)(27,37)(29,39)(30,32)(31,36)(33,35)(38,40);(1,3,4,7)(2,6,5,8)(9,14,16,10)(11,17,15,20)(12,19,13,18)(21,23,22,24)(25,29,31,37)(26,35,32,38)(27,28,39,36)(30,40,34,33);(9,15,16,11)(10,17,14,20)(12,22,13,21)(18,23,19,24)(25,30,31,34)(26,36,32,28)(27,38,39,35)(29,40,37,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=hi,e^2=g,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, 8412233);