#8892857

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

4347 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)(12,17)(13,22)(14,15)(18,19)(20,23)(21,24)(25,26)(27,36)(28,39)(29,40)(30,31)(32,35)(33,38)(34,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27,32,38)(26,33,35,36)(28,37,30,29)(31,34,39,40);(2,5)(6,8)(10,15)(13,17)(16,23)(19,24)(25,28,32,30)(26,31,35,39)(27,37,38,29)(33,34,36,40);(3,7)(6,8)(9,11,14,20)(10,16,15,23)(12,21,22,18)(13,19,17,24)(25,29)(26,34)(27,28)(30,38)(31,36)(32,37)(33,39)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,30,32,28)(26,31,35,39)(27,37,38,29)(33,40,36,34);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,31)(26,28)(27,34)(29,36)(30,35)(32,39)(33,37)(38,40)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800053);