#8936722

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

4554 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=h,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, 8843918);