#8276378

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

4441 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,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, 8183574);