#8346827

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

4604 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)(12,13)(14,15)(16,19)(17,22)(20,21)(23,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,7)(4,5)(6,8)(10,15)(11,19)(12,22)(21,24)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(2,5)(6,7)(10,15)(13,17)(16,18)(21,24)(25,27,30,28)(26,32,34,31)(29,36,37,35)(33,38,40,39);(3,8)(6,7)(9,11,14,19)(10,16,15,18)(12,23,22,20)(13,21,17,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,15)(14,22)(16,21)(18,24)(19,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,12,15,22)(11,21,19,24)(16,23,18,20)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=g,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,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, 8254023);