#8898064

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

4459 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)(3,7)(4,5)(6,8)(25,26,32,34)(27,28,30,29)(31,39,40,33)(35,37,38,36);(2,5)(6,7)(9,10,14,15)(11,16,21,24)(12,17,22,20)(13,18,23,19)(25,27)(26,28)(29,34)(30,32)(31,35)(33,36)(37,39)(38,40);(1,3,4,8)(2,6,5,7)(10,15)(13,23)(16,24)(17,20)(25,28,32,29)(26,27,34,30)(31,37,40,36)(33,38,39,35);(2,5)(6,7)(9,11)(10,16)(12,18)(13,20)(14,21)(15,24)(17,23)(19,22)(25,29,32,28)(26,27,34,30)(31,37,40,36)(33,35,39,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,30,32,27)(26,29,34,28)(31,38,40,35)(33,37,39,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,31,32,40)(26,33,34,39)(27,35,30,38)(28,36,29,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805260);