#10463562

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

4744 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10370758);