#8899244

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	#35508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806440);