#8943454

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4237 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850650);