#10020649

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,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, 9927845);