#8704085

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8611281);