#8648962

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

4582 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8556158);