#8096060

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

4505 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)(3,8)(4,6)(5,7)(9,10,14,18)(11,13,17,15)(12,16,20,23)(19,21,24,22)(25,26,31,35)(27,29,30,32)(28,34,39,33)(36,40,37,38);(1,3,5,4)(2,6,7,8)(12,20)(16,23)(19,24)(21,22)(25,27,31,30)(26,32,35,29)(28,37,39,36)(33,38,34,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,40)(34,35);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,29,31,32)(26,27,35,30)(28,38,39,40)(33,36,34,37);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,28)(26,34)(27,37)(29,38)(30,36)(31,39)(32,40)(33,35);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,23)(20,22)(25,30,31,27)(26,32,35,29)(28,37,39,36)(33,40,34,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003256);