#10046990

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

4454 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)(3,5)(4,7)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,26)(27,29)(28,34)(30,32)(31,33)(35,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,18)(10,15,16,13)(12,20,21,19)(17,23,24,22)(25,27,31,30)(26,32,33,29)(28,36,37,35)(34,38,40,39);(1,4,6,5)(2,7,8,3)(10,16)(11,18)(12,21)(22,23)(26,33)(28,37)(29,32)(35,36);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,23)(20,21)(22,24)(25,28,31,37)(26,34,33,40)(27,35,30,36)(29,38,32,39);(1,2)(3,4)(5,7)(6,8)(9,12,14,21)(10,17,16,24)(11,19,18,20)(13,23,15,22)(25,29,31,32)(26,27,33,30)(28,38,37,39)(34,35,40,36);(1,5,6,4)(2,3,8,7)(9,13)(10,11)(12,22)(14,15)(16,18)(17,20)(19,24)(21,23)(25,30,31,27)(26,29,33,32)(28,36,37,35)(34,38,40,39)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954186);