#8926451

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

4357 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)(9,10,15,17)(11,20,12,16)(13,18,19,14)(21,22,23,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(2,5)(3,8)(9,11,15,12)(10,16,17,20)(13,21,19,23)(14,22,18,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3)(2,6)(4,5)(7,8)(10,17)(14,18)(16,20)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,7,6)(2,3,5,8)(9,12,15,11)(10,16,17,20)(13,21,19,23)(14,24,18,22);(1,5,7,2)(3,4,8,6)(9,13,15,19)(10,18,17,14)(11,21,12,23)(16,24,20,22)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,6,7,4)(2,8,5,3)(9,14,15,18)(10,19,17,13)(11,22,12,24)(16,23,20,21)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833647);