#8936118

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

4682 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)(11,17)(12,16)(13,19)(14,15)(18,24)(20,22)(21,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,20)(14,19)(15,17)(16,24)(18,21)(22,23)(25,27,30,33)(26,31,29,35)(28,34,38,36)(32,39,37,40);(2,5)(3,7)(10,15)(12,21)(13,17)(18,20)(26,29)(27,33)(32,37)(34,36);(3,7)(6,8)(9,11)(10,13)(12,20)(14,19)(15,17)(16,24)(18,21)(22,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,16,15,23)(11,18,19,20)(13,22,17,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,22)(14,17)(15,19)(16,18)(20,23)(21,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)

Polycyclic presentation:

2^9|b^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, 8843314);