#8843874

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

4565 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,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, 8751070);