#8905172

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

4277 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812368);