#8362184

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	#47870	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=hi,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,e^d=egi

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, 8269380);