#8358609

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

4456 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=i,e^2=gh,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, 8265805);