#8341802

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

4349 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8248998);