#8108715

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 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)(9,10,14,16)(11,17,18,12)(13,21,22,15)(19,20,24,23)(26,28)(30,32)(33,36)(37,39);(1,2,4,6)(3,5,8,7)(11,18)(12,17)(13,22)(15,21)(25,26,31,28)(27,32,35,30)(29,36,38,33)(34,37,40,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,18)(15,20)(16,17)(21,23)(22,24)(25,27,31,35)(26,32,28,30)(29,34,38,40)(33,39,36,37);(1,3,4,8)(2,5,6,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28,31,26)(27,30,35,32)(29,36,38,33)(34,37,40,39);(9,13,14,22)(10,15,16,21)(11,19,18,24)(12,20,17,23)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,30)(26,27)(28,35)(29,37)(31,32)(33,34)(36,40)(38,39)

Polycyclic presentation:

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

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, 8015911);