#10521327

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

4382 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)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,26)(27,32)(28,38)(29,33)(30,39)(31,35)(34,36)(37,40);(2,5)(6,7)(9,11,15,13)(10,16,18,12)(14,19,22,21)(17,23,24,20)(25,27,31,37)(26,32,35,40)(28,36,29,30)(33,39,38,34);(1,2)(3,6)(4,5)(7,8)(25,28,31,29)(26,33,35,38)(27,36,37,30)(32,39,40,34);(1,3)(2,6)(4,8)(5,7)(9,12,15,16)(10,11,18,13)(14,20,22,23)(17,19,24,21)(25,29,31,28)(26,33,35,38)(27,30,37,36)(32,39,40,34);(9,13,15,11)(10,16,18,12)(14,19,22,21)(17,20,24,23)(25,26)(27,32)(28,33)(29,38)(30,34)(31,35)(36,39)(37,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30)(26,34)(27,28)(29,37)(31,36)(32,33)(35,39)(38,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=ghi,e^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, 10428523);