#8947338

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

4303 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,e^2=gh,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, 8854534);