#8915204

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8822400);