#8898346

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	#46397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^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, 8805542);