#10196162

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

4581 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,19,17)(12,15,21,24)(18,22,20,23)(25,26,30,31)(27,28,32,36)(29,37,39,33)(34,38,40,35);(1,3,7,6)(2,4,8,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,38)(39,40);(1,4,7,5)(2,3,8,6)(26,31)(27,32)(33,37)(34,40);(2,8)(4,5)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,38)(39,40);(1,5,7,4)(2,3,8,6)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28)(26,32)(27,31)(29,38)(30,36)(33,34)(35,39)(37,40);(1,6,7,3)(2,5,8,4)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,30,39)(26,33,31,37)(27,34,32,40)(28,35,36,38)

Polycyclic presentation:

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

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, 10103358);