#8936692

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

4455 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,f^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, 8843888);