#8936729

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

4650 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)(11,18)(12,20)(13,17)(14,15)(16,22)(19,23)(21,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,17)(10,13,14,18)(12,19,22,24)(16,23,20,21)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(2,5)(3,7)(10,14)(13,18)(16,20)(21,23)(25,27,30,28)(26,32,34,31)(29,36,37,35)(33,38,40,39);(3,7)(6,8)(9,12)(10,16)(11,19)(13,23)(14,20)(15,22)(17,24)(18,21)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(1,3)(2,6)(4,7)(5,8)(9,13,15,18)(10,17,14,11)(12,21,22,23)(16,19,20,24)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,18,17,13)(12,20,22,16)(19,21,24,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=gh,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, 8843925);