#8198194

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	#56401	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4369 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8105390);