#8901273

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
prime radical	#51204	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=ghi,f^2=g,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, 8808469);