#9206639

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	#47679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 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:

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

Polycyclic presentation:

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

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, 9113835);