#8260652

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

4618 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=h,e^2=hi,f^2=hi,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, 8167848);