#8346707

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^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, 8253903);