#10519385

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

4501 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)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(10,14)(15,21)(16,22)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,3,6,5)(2,7,4,8)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,6,5)(2,8,4,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,4,6,2)(3,8,5,7)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,5,6,3)(2,8,4,7)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=ghi,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, 10426581);