#8904604

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	#46983	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8811800);