#8417081

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

4407 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324277);