#8850022

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

4693 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)(9,10,14,15)(11,16,19,13)(12,20,23,17)(18,22,21,24)(26,28)(27,32)(30,38)(37,40);(1,2,4,6)(3,5,7,8)(10,15)(13,16)(17,20)(22,24)(25,26,31,28)(27,34,32,29)(30,33,38,36)(35,40,39,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,16,15,13)(12,18,23,21)(17,24,20,22)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,40)(38,39);(9,11)(10,13)(12,21)(14,19)(15,16)(17,24)(18,23)(20,22)(25,28,31,26)(27,34,32,29)(30,36,38,33)(35,40,39,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,23)(10,17,15,20)(11,18,19,21)(13,22,16,24)(25,29,31,34)(26,32,28,27)(30,37,38,40)(33,39,36,35);(1,3)(2,5)(4,7)(6,8)(9,13)(10,11)(12,22)(14,16)(15,19)(17,18)(20,21)(23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=i,e^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, 8757218);