#9163971

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×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9071167);