#8091732

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

4529 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7998928);