#8934751

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

4430 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)(3,5)(4,8)(6,7)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,34)(27,31,32,28)(29,33,38,40)(35,39,37,36);(1,3)(2,5)(4,6)(7,8)(9,10,13,14)(11,15,19,17)(12,16,21,20)(18,22,24,23)(25,27,30,32)(26,28,34,31)(29,35,38,37)(33,36,40,39);(1,2)(3,8)(4,5)(6,7)(10,14)(12,21)(15,17)(18,24)(25,27,30,32)(26,31,34,28)(29,37,38,35)(33,36,40,39);(2,7)(3,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,40)(37,38);(1,4,6,3)(2,5,7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,5)(2,3)(4,7)(6,8)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841947);