#8132498

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

4273 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8039694);