#8127629

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

4502 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,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, 8034825);