#8285550

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

4458 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)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,31,33)(27,35,29,32)(28,38,39,34)(30,36,40,37);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,29)(26,32,33,35)(28,36,39,37)(30,34,40,38);(1,2)(3,5)(4,6)(7,8)(12,20)(15,22)(18,24)(21,23)(26,33)(30,40)(32,35)(34,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,31,39)(26,34,33,38)(27,37,29,36)(30,35,40,32);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,31,27)(26,35,33,32)(28,37,39,36)(30,38,40,34);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30)(26,36)(27,34)(28,35)(29,38)(31,40)(32,39)(33,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=h,e^2=h,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, 8192746);