#10195881

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

4642 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=h,e^2=gh,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, 10103077);