#8227451

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

4543 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8134647);