#8057909

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

4189 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:

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

Polycyclic presentation:

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

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, 7965105);