#8810206

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

4319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8717402);