#10419704

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

4385 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)(25,26,32,35)(27,30,34,40)(28,37,39,33)(29,31,38,36);(2,5)(6,7)(9,11,14,12)(10,15,18,16)(13,19,21,20)(17,22,24,23)(25,27,32,34)(26,30,35,40)(28,36,39,31)(29,33,38,37);(1,2)(3,6)(4,5)(7,8)(25,28,32,39)(26,33,35,37)(27,36,34,31)(29,30,38,40);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,16,18,15)(13,19,21,20)(17,22,24,23)(25,29)(26,31)(27,33)(28,30)(32,38)(34,37)(35,36)(39,40);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,30)(26,34)(27,35)(28,38)(29,39)(31,33)(32,40)(36,37);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,31)(26,29)(27,28)(30,33)(32,36)(34,39)(35,38)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,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, 10326900);