#10482487

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
prime radical	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gi,e^2=g,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, 10389683);