#10042571

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	#44234	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 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)(3,4)(5,6)(7,8)(11,18)(16,19)(20,23)(21,24)(25,26,31,29)(27,34,30,32)(28,33,37,36)(35,40,38,39);(1,3,6,7)(2,4,5,8)(9,10,14,15)(11,19,18,16)(12,13,17,22)(20,24,23,21)(25,27,31,30)(26,32,29,34)(28,35,37,38)(33,39,36,40);(1,4,6,8)(2,7,5,3)(10,15)(12,17)(16,19)(20,23)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,2,6,5)(3,8,7,4)(9,11,14,18)(10,16,15,19)(12,20,17,23)(13,21,22,24)(25,29,31,26)(27,32,30,34)(28,36,37,33)(35,39,38,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,13)(11,20)(14,17)(15,22)(16,21)(18,23)(19,24)(25,30,31,27)(26,32,29,34)(28,35,37,38)(33,40,36,39);(1,5,6,2)(3,8,7,4)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,24)(19,20)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=gi,e^2=g,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, 9949767);