#9139460

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	#43110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9046656);