#10422620

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

4719 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22)(25,26)(27,33)(28,30)(29,36)(31,35)(32,38)(34,39)(37,40);(2,5)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(26,28)(31,33)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,40)(34,38)(36,39);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10329816);