#8215348

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=gh,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, 8122544);