#10529222

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

4392 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=g,f^2=gi,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, 10436418);