#9065817

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	#35752	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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973013);