#8839122

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

4694 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746318);