#8288382

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,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, 8195578);