#8898521

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

4435 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=h,e^2=g,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, 8805717);