#8148408

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8055604);