#8231417

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	#45608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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)(12,14)(13,17)(15,19)(18,23)(20,24)(21,22)(25,26)(27,34)(28,36)(29,31)(30,32)(33,37)(35,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,23)(10,16,19,18)(12,21,13,20)(14,22,17,24)(25,27,31,30)(26,32,29,34)(28,35,37,38)(33,39,36,40);(2,6)(4,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,22,18,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,22,18,24)(25,26,31,29)(27,34,30,32)(28,33,37,36)(35,40,38,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,18)(12,20)(13,21)(14,22)(15,23)(16,19)(17,24)(25,29,31,26)(27,32,30,34)(28,36,37,33)(35,39,38,40);(1,4,5,7)(2,3,6,8)(9,14,15,17)(10,12,19,13)(11,22,23,24)(16,21,18,20)(25,30,31,27)(26,32,29,34)(28,35,37,38)(33,40,36,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8138613);