#10519381

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

4606 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)(11,16)(12,17)(13,18)(14,15)(19,20)(21,23)(22,24)(25,26,31,35)(27,34,38,32)(28,30,29,33)(36,37,39,40);(10,15)(11,18)(17,23)(19,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,38)(33,40)(34,35);(1,3,6,5)(2,7,4,8)(25,28,31,29)(26,33,35,30)(27,36,38,39)(32,37,34,40);(1,3,6,5)(2,8,4,7)(9,11,14,18)(10,16,15,13)(12,19,21,24)(17,20,23,22)(25,29,31,28)(26,30,35,33)(27,36,38,39)(32,37,34,40);(1,4,6,2)(3,8,5,7)(9,12)(10,17)(11,19)(13,22)(14,21)(15,23)(16,20)(18,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,5,6,3)(2,8,4,7)(9,13,14,16)(10,18,15,11)(12,20,21,22)(17,19,23,24)(25,30)(26,28)(27,37)(29,35)(31,33)(32,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=ghi,e^2=hi,f^2=gh,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, 10426577);