#10040771

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	#44783	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,26,31,29)(27,32,35,39)(28,38,37,33)(30,40,36,34);(1,3,5,6)(2,4,7,8)(10,14)(11,17)(16,19)(18,20)(25,27,31,35)(26,32,29,39)(28,30,37,36)(33,34,38,40);(1,4)(2,6)(3,7)(5,8)(10,14)(11,17)(16,19)(18,20)(26,29)(27,35)(28,37)(34,40);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,28,31,37)(26,33,29,38)(27,36,35,30)(32,40,39,34);(1,2)(3,4)(5,7)(6,8)(9,11,13,17)(10,15,14,22)(12,20,21,18)(16,24,19,23)(25,29,31,26)(27,32,35,39)(28,38,37,33)(30,34,36,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,30,31,36)(26,34,29,40)(27,37,35,28)(32,38,39,33)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,e^2=h,f^2=h,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, 9947967);