#10216842

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

4439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gi,e^2=ghi,f^2=ghi,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, 10124038);