#7714471

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621667);