#8188172

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

4460 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,20)(14,16)(15,19)(17,22)(18,23)(21,24)(25,26)(27,37)(28,34)(29,39)(30,35)(31,33)(32,36)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,27,32,31)(26,33,36,37)(28,38,29,35)(30,39,40,34);(1,2,4,6)(3,5,8,7)(10,16)(13,15)(17,20)(21,23)(25,28,32,29)(26,34,36,39)(27,38,31,35)(30,37,40,33);(9,11)(10,15)(12,18)(13,16)(14,19)(17,23)(20,21)(22,24)(25,29,32,28)(26,34,36,39)(27,38,31,35)(30,33,40,37);(1,3)(2,5)(4,8)(6,7)(9,12,14,22)(10,17,16,20)(11,18,19,24)(13,21,15,23)(25,30,32,40)(26,35,36,38)(27,39,31,34)(28,37,29,33);(1,2)(3,5)(4,6)(7,8)(9,13,14,15)(10,11,16,19)(12,21,22,23)(17,18,20,24)(25,31,32,27)(26,33,36,37)(28,38,29,35)(30,34,40,39)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8095368);