#8201751

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

4872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,f^2=g,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, 8108947);