#8107323

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	#40661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4546 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,22)(15,17)(18,24)(19,21)(20,23)(27,33)(30,36)(34,39)(37,40);(2,6)(3,7)(9,11,15,20)(10,16,17,23)(12,14,18,21)(13,22,24,19)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,3,5,7)(2,4,6,8)(10,17)(11,20)(12,18)(19,22)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,4)(2,7)(3,6)(5,8)(9,12,15,18)(10,13,17,24)(11,21,20,14)(16,19,23,22)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(9,13)(10,18)(11,22)(12,17)(14,23)(15,24)(16,21)(19,20)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,14,15,21)(10,19,17,22)(11,18,20,12)(13,23,24,16)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014519);