#8850038

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

4338 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)(9,10,15,12)(11,18,16,13)(14,20,22,17)(19,21,23,24)(26,31)(27,29)(33,36)(35,37);(1,2,4,6)(3,5,7,8)(9,10,15,12)(11,18,16,13)(14,20,22,17)(19,21,23,24)(25,26,30,31)(27,32,29,34)(28,36,38,33)(35,40,37,39);(1,3)(2,5)(4,7)(6,8)(9,11,15,16)(10,13,12,18)(14,19,22,23)(17,21,20,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(9,12,15,10)(11,13,16,18)(14,20,22,17)(19,21,23,24)(25,27,30,29)(26,32,31,34)(28,35,38,37)(33,39,36,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,12)(14,21)(15,18)(17,23)(19,20)(22,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,27)(26,34,31,32)(28,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8757234);