#8944536

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
prime radical	#40585	C₂
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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,f^2=g,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, 8851732);