#9015516

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

4642 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)(3,8)(4,7)(5,6)(10,15)(11,18)(17,23)(19,21)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(3,4)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,24)(20,23)(21,22)(26,28)(31,33)(32,36)(39,40);(1,3)(2,7)(4,6)(5,8)(11,18)(12,20)(13,16)(17,23)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(3,4)(7,8)(9,11,14,18)(10,16,15,13)(12,21,20,19)(17,22,23,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,4,6,3)(2,7,5,8)(9,12,14,20)(10,17,15,23)(11,19,18,21)(13,22,16,24)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(1,5,6,2)(3,7,4,8)(9,13,14,16)(10,18,15,11)(12,22,20,24)(17,21,23,19)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8922712);