#10040773

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	#43504	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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,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, 9947969);