#10040818

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	#41316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4236 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)(3,7)(4,6)(5,8)(9,10,15,13)(11,19,21,16)(12,22,23,17)(14,24,20,18)(25,26)(27,33)(28,30)(29,39)(31,34)(32,35)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,13,19)(12,20,23,14)(17,18,22,24)(25,27)(26,33)(28,38)(29,32)(30,40)(31,36)(34,37)(35,39);(1,2)(3,5)(4,6)(7,8)(9,12,15,23)(10,17,13,22)(11,14,21,20)(16,24,19,18)(25,28,32,31)(26,34,35,30)(27,36,29,38)(33,40,39,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,20)(13,22)(14,21)(15,23)(16,18)(19,24)(25,29,32,27)(26,33,35,39)(28,36,31,38)(30,40,34,37);(9,13,15,10)(11,16,21,19)(12,22,23,17)(14,24,20,18)(25,30,32,34)(26,28,35,31)(27,37,29,40)(33,36,39,38);(9,14,15,20)(10,18,13,24)(11,12,21,23)(16,22,19,17)(25,31,32,28)(26,30,35,34)(27,36,29,38)(33,40,39,37)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,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, 9948014);