#10040867

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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)(9,10,14,18)(11,19,20,15)(12,16,22,21)(13,23,24,17)(25,26)(27,35)(28,38)(29,32)(30,34)(31,33)(36,39)(37,40);(1,2,4,6)(3,5,8,7)(11,20)(12,22)(15,19)(16,21)(25,27,32,31)(26,33,29,35)(28,37,30,36)(34,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,18,19)(12,23,22,17)(13,21,24,16)(25,28,32,30)(26,34,29,38)(27,36,31,37)(33,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,17)(13,21)(14,20)(16,24)(18,19)(22,23)(25,29,32,26)(27,35,31,33)(28,34,30,38)(36,39,37,40);(9,12,14,22)(10,16,18,21)(11,17,20,23)(13,15,24,19)(25,30,32,28)(26,34,29,38)(27,37,31,36)(33,39,35,40);(9,13,14,24)(10,17,18,23)(11,21,20,16)(12,19,22,15)(25,31,32,27)(26,35,29,33)(28,37,30,36)(34,40,38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,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, 9948063);