#8374807

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8282003);