#8592259

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

4595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^2=gh,f^2=g,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, 8499455);