#8075345

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

4297 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982541);