#10143529

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

4661 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=ghi,e^2=h,f^2=gh,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, 10050725);