#10171836

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	#43396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10079032);