#10480767

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

4611 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,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, 10387963);