#10041423

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

4713 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)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,29,30)(27,31,34,33)(28,36,38,32)(35,40,37,39);(1,3,5,4)(2,6,7,8)(12,19)(15,17)(20,24)(22,23)(25,26,29,30)(27,33,34,31)(28,32,38,36)(35,40,37,39);(1,2)(3,6)(4,8)(5,7)(10,11)(15,17)(16,18)(22,23)(26,30)(27,34)(32,36)(35,37);(1,2)(3,8)(4,6)(5,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,4,5,3)(2,6,7,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,31)(28,37)(29,34)(30,33)(32,40)(35,38)(36,39);(1,2)(3,6)(4,8)(5,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=hi,e^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, 9948619);