#8942816

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

4484 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)(11,12)(13,21)(14,15)(16,22)(17,18)(19,24)(20,23)(25,26,31,33)(27,29,35,32)(28,30,39,34)(36,38,40,37);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,19)(14,17)(15,18)(16,20)(21,24)(22,23)(25,27,31,35)(26,32,33,29)(28,36,39,40)(30,37,34,38);(1,2,4,6)(3,7,8,5)(10,15)(11,17)(13,22)(20,24)(26,33)(29,32)(30,34)(37,38);(3,8)(5,7)(9,12)(10,11)(13,20)(14,18)(15,17)(16,19)(21,23)(22,24)(25,28)(26,34)(27,36)(29,37)(30,33)(31,39)(32,38)(35,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,17)(13,16)(14,15)(19,24)(20,23)(21,22)(25,29)(26,35)(27,33)(28,38)(30,40)(31,32)(34,36)(37,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30,31,34)(26,28,33,39)(27,37,35,38)(29,40,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,f^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, 8850012);