#10040777

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	#41949	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9947973);