#8968070

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

4372 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8875266);