#8126031

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

4621 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)(9,10,13,15)(11,14,17,22)(12,19,20,16)(18,24,23,21)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,14,15,22)(12,18,20,23)(16,21,19,24)(25,27,31,30)(26,32,29,34)(28,37,38,35)(33,40,36,39);(2,5)(6,8)(10,15)(11,17)(16,19)(18,23)(26,29)(28,38)(32,34)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28)(26,33)(27,35)(29,36)(30,37)(31,38)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,21,22,24)(25,30,31,27)(26,34,29,32)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=g,f^2=h,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, 8033227);