#9726996

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	#39092	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,31,33)(27,36,38,32)(28,34,39,29)(30,37,40,35);(1,3,6,7)(2,4,5,8)(11,17)(15,21)(18,23)(22,24)(25,27,31,38)(26,32,33,36)(28,37,39,35)(29,40,34,30);(1,2,6,5)(3,8,7,4)(10,14)(15,21)(16,19)(22,24)(26,33)(28,39)(32,36)(35,37);(1,4)(2,7)(3,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28)(26,34)(27,37)(29,33)(30,32)(31,39)(35,38)(36,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,31,34)(26,28,33,39)(27,30,38,40)(32,35,36,37);(1,5,6,2)(3,8,7,4)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,35)(27,29)(28,32)(31,40)(33,37)(34,38)(36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9634192);