#8158735

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

4310 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)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,5,4)(2,6,7,8)(11,17)(14,20)(18,23)(21,24)(26,31)(27,32)(33,38)(34,39);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27,30,32)(26,28,31,36)(29,34,37,39)(33,35,38,40);(2,7)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24);(1,4,5,3)(2,8,7,6)(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065931);