#10463931

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

4343 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)(9,10,14,17)(11,15,20,13)(12,16,22,24)(18,23,19,21)(26,31)(27,32)(29,37)(35,38);(1,3,6,5)(2,7,4,8)(9,11,14,20)(10,15,17,13)(12,19,22,18)(16,21,24,23)(26,31)(27,32)(33,39)(34,40);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,26,30,31)(27,28,32,36)(29,33,37,39)(34,35,40,38);(1,2,6,4)(3,8,5,7)(9,10)(11,13)(12,16)(14,17)(15,20)(18,21)(19,23)(22,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,19)(13,23)(14,22)(15,21)(17,24)(18,20)(25,28)(26,32)(27,31)(29,38)(30,36)(33,34)(35,37)(39,40);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,20)(19,24)(22,23)(25,29)(26,33)(27,34)(28,35)(30,37)(31,39)(32,40)(36,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,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^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371127);