#10058508

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

4469 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

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, 9965704);