#9058255

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

4523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965451);