#10075902

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

4789 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,30,33)(27,28,31,34)(29,32,38,40)(35,36,39,37);(1,3,6,7)(2,4,5,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,4,6,8)(2,7,5,3)(10,11)(15,17)(16,18)(22,23)(27,31)(28,34)(35,39)(36,37);(1,2,6,5)(3,8,7,4)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(1,5,6,2)(3,8,7,4)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9983098);