#8390895

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4519 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)(11,18)(12,22)(13,17)(14,15)(16,21)(19,24)(20,23)(25,26)(27,28)(29,30)(31,33)(32,34)(35,36)(37,38)(39,40);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,18)(19,23)(25,27)(26,29)(28,34)(30,32)(31,35)(33,37)(36,40)(38,39);(2,5)(6,7)(10,15)(13,22)(16,18)(20,24)(25,28,32,29)(26,27,34,30)(31,37,39,36)(33,38,40,35);(1,3,4,8)(2,6,5,7)(9,11,14,21)(10,16,15,18)(12,23,17,19)(13,20,22,24)(25,29,32,28)(26,27,34,30)(31,37,39,36)(33,35,40,38);(1,3)(2,6)(4,8)(5,7)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,24,18,20)(25,30,32,27)(26,29,34,28)(31,38,39,35)(33,37,40,36);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,23)(21,24)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,39)(34,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8298091);