#9023632

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

4592 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)(3,7)(4,5)(6,8)(9,10,15,13)(11,12,19,16)(14,21,22,17)(18,23,24,20)(25,26,31,35)(27,32,30,34)(28,29,33,39)(36,37,38,40);(3,8)(6,7)(9,11,15,19)(10,16,13,12)(14,18,22,24)(17,23,21,20)(25,27,31,30)(26,32,35,34)(28,36,33,38)(29,37,39,40);(1,3,5,8)(2,6,4,7)(10,13)(12,16)(14,22)(18,24)(27,30)(28,33)(29,39)(32,34);(3,8)(6,7)(9,12)(10,11)(13,19)(14,20)(15,16)(17,18)(21,24)(22,23)(25,28)(26,29)(27,36)(30,38)(31,33)(32,37)(34,40)(35,39);(1,3)(2,7)(4,6)(5,8)(9,13,15,10)(11,16,19,12)(14,21,22,17)(18,23,24,20)(25,29)(26,33)(27,37)(28,35)(30,40)(31,39)(32,38)(34,36);(1,4,5,2)(3,6,8,7)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,34,35,32)(28,38,33,36)(29,40,39,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,e^2=g,f^2=i,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, 8930828);