#8122929

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

4472 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)(11,18)(12,16)(13,23)(14,24)(15,17)(19,22)(20,21)(25,26)(27,33)(28,29)(30,32)(31,35)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(11,12)(16,18)(19,21)(20,22)(25,27,31,30)(26,32,35,33)(28,36,39,38)(29,37,34,40);(2,5)(6,8)(9,11,15,12)(10,16,17,18)(13,21,24,19)(14,20,23,22)(25,27,31,30)(26,33,35,32)(28,36,39,38)(29,40,34,37);(9,12,15,11)(10,16,17,18)(13,21,24,19)(14,22,23,20)(25,28)(26,34)(27,36)(29,35)(30,38)(31,39)(32,40)(33,37);(1,2)(3,6)(4,5)(7,8)(9,13)(10,14)(11,19)(12,21)(15,24)(16,22)(17,23)(18,20)(25,29,31,34)(26,28,35,39)(27,37,30,40)(32,36,33,38);(1,3,4,7)(2,6,5,8)(9,14)(10,13)(11,20)(12,22)(15,23)(16,21)(17,24)(18,19)(25,30,31,27)(26,33,35,32)(28,38,39,36)(29,40,34,37)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8030125);