#8288392

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

4624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8195588);