#8323223

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

4354 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,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=dh,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, 8230419);