#8119618

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

4486 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026814);