#8527388

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

4577 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434584);