#8126051

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

4379 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)(9,10,15,19)(11,20,22,16)(12,13,23,17)(14,24,21,18)(26,30)(31,36)(32,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,14)(13,18)(15,22)(17,24)(19,20)(21,23)(27,33)(31,36)(34,39)(38,40);(2,5)(6,8)(9,12,15,23)(10,17,19,13)(11,14,22,21)(16,24,20,18)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,19)(14,22)(15,23)(16,18)(20,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,12,19,23)(11,18,22,24)(14,20,21,16)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,14,15,21)(10,18,19,24)(11,12,22,23)(13,20,17,16)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,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, 8033247);