#8943827

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×#534	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^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, 8851023);