#8944583

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

4309 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)(9,10,15,13)(11,17,12,16)(14,21,22,18)(19,23,20,24);(3,7)(5,8)(10,13)(16,17)(18,21)(23,24)(25,26,30,35)(27,33,36,31)(28,32,38,39)(29,34,40,37);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,13,17)(14,20,22,19)(18,24,21,23)(25,27,30,36)(26,31,35,33)(28,34,38,37)(29,39,40,32);(9,12,15,11)(10,17,13,16)(14,20,22,19)(18,24,21,23)(25,28)(26,32)(27,34)(29,31)(30,38)(33,40)(35,39)(36,37);(1,3)(2,5)(4,7)(6,8)(9,13,15,10)(11,16,12,17)(14,21,22,18)(19,23,20,24)(25,27)(26,33)(28,37)(29,32)(30,36)(31,35)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29)(26,34)(27,32)(28,31)(30,40)(33,38)(35,37)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8851779);