#10040841

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	#52553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=ghi,e^2=h,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, 9948037);