#10040722

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,20)(13,18)(14,22)(15,19)(17,23)(21,24)(25,26,31,29)(27,34,30,32)(28,33,38,37)(35,40,36,39);(1,3)(2,4)(5,6)(7,8)(9,11,15,14)(10,16,19,22)(12,21,23,18)(13,20,24,17)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,2)(3,4)(5,7)(6,8)(9,12,15,23)(10,17,19,20)(11,18,14,21)(13,16,24,22)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,2)(3,8)(4,6)(5,7)(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,24)(19,20)(25,28,31,38)(26,33,29,37)(27,35,30,36)(32,39,34,40);(1,4,5,8)(2,6,7,3)(9,13)(10,18)(11,17)(12,22)(14,20)(15,24)(16,23)(19,21)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,16,19,22)(12,21,23,18)(13,17,24,20)(25,30,31,27)(26,34,29,32)(28,36,38,35)(33,40,37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 9947918);