#8132814

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

4454 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)(11,13)(12,15)(14,20)(16,17)(18,22)(19,23)(21,24)(27,34)(29,32)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(25,26,31,30)(27,32,34,29)(28,36,38,33)(35,37,40,39);(2,5)(6,8)(10,12)(13,16)(18,20)(21,23)(25,27,31,34)(26,32,30,29)(28,35,38,40)(33,39,36,37);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,16,17,13)(14,20,22,18)(19,23,24,21)(25,28,31,38)(26,33,30,36)(27,35,34,40)(29,37,32,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,16)(10,17,12,11)(14,21,22,23)(18,24,20,19)(25,29)(26,27)(28,37)(30,34)(31,32)(33,35)(36,40)(38,39);(1,3,4,7)(2,6,5,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30,31,26)(27,29,34,32)(28,36,38,33)(35,37,40,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8040010);