#8799407

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706603);