#8366603

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

4291 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,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,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, 8273799);