#10075906

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

4502 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,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, 9983102);