#8131279

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8038475);