#8108123

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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=g,e^2=g,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^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015319);