#8399419

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

4570 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)(6,8)(9,10,15,17)(11,16,20,23)(12,18,22,14)(13,21,24,19)(26,31)(27,28)(33,38)(34,36);(3,7)(6,8)(9,11)(10,16)(12,21)(13,14)(15,20)(17,23)(18,24)(19,22)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,17)(11,20)(14,18)(19,21)(25,27,30,28)(26,32,31,35)(29,36,37,34)(33,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,21)(13,23)(14,17)(15,22)(16,24)(19,20)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,19)(11,18)(12,16)(14,20)(15,24)(17,21)(22,23)(25,28,30,27)(26,32,31,35)(29,36,37,34)(33,39,38,40);(9,14,15,18)(10,12,17,22)(11,13,20,24)(16,21,23,19)(25,29,30,37)(26,33,31,38)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=g,f^2=gi,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, 8306615);