#10040584

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	#44593	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,18)(14,16)(15,17)(19,22)(20,23)(21,24)(25,26,31,33)(27,32,37,40)(28,38,39,34)(29,35,36,30);(1,3,5,4)(2,6,7,8)(9,11,15,14)(10,16,17,13)(12,20,21,19)(18,22,24,23)(25,27)(26,32)(28,30)(29,38)(31,37)(33,40)(34,36)(35,39);(1,2)(3,6)(4,8)(5,7)(10,17)(12,21)(13,16)(19,20)(26,33)(27,37)(28,39)(29,36);(1,2)(3,8)(4,6)(5,7)(9,12,15,21)(10,18,17,24)(11,19,14,20)(13,22,16,23)(25,28)(26,34)(27,35)(29,32)(30,37)(31,39)(33,38)(36,40);(1,4,5,3)(2,6,7,8)(9,13,15,16)(10,11,17,14)(12,22,21,23)(18,19,24,20)(25,29)(26,35)(27,34)(28,32)(30,33)(31,36)(37,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,11)(10,13,17,16)(12,20,21,19)(18,22,24,23)(25,30,31,35)(26,36,33,29)(27,28,37,39)(32,34,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 9947780);