#10040670

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

4490 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)(3,8)(4,6)(5,7)(9,10,14,16)(11,17,18,12)(13,21,23,15)(19,20,24,22)(26,29)(31,33)(32,37)(38,39);(1,3,5,4)(2,6,7,8)(11,18)(12,17)(19,24)(20,22)(28,36)(32,37)(34,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,11,14,18)(10,12,16,17)(13,19,23,24)(15,20,21,22)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,22)(23,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,22)(14,17)(15,24)(16,18)(19,21)(20,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,2)(3,6)(4,8)(5,7)(9,13,14,23)(10,15,16,21)(11,19,18,24)(12,20,17,22)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=h,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, 9947866);