#8197927

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4465 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,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, 8105123);