#8346897

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

4526 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)(11,17)(12,13)(14,18)(15,16)(19,24)(20,21)(22,23)(25,26,31,34)(27,32,36,40)(28,37,35,33)(29,39,38,30);(1,2)(3,7)(4,5)(6,8)(10,16)(13,17)(18,23)(21,24)(25,27,31,36)(26,32,34,40)(28,38,35,29)(30,33,39,37);(2,5)(6,7)(9,11,15,12)(10,17,16,13)(14,20,22,19)(18,21,23,24)(25,28,31,35)(26,33,34,37)(27,29,36,38)(30,40,39,32);(3,8)(6,7)(9,12,15,11)(10,17,16,13)(14,20,22,19)(18,24,23,21)(25,27)(26,32)(28,29)(30,33)(31,36)(34,40)(35,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,17)(10,12,16,11)(14,21,22,24)(18,20,23,19)(25,29)(26,30)(27,35)(28,36)(31,38)(32,37)(33,40)(34,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30)(26,29)(27,33)(28,32)(31,39)(34,38)(35,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,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,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, 8254093);