#9713384

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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)(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,36)(28,34)(29,35)(30,32)(31,33)(37,40)(38,39);(1,3)(2,4)(5,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,27,31,30)(26,32,33,36)(28,37,35,39)(29,40,34,38);(1,2,7,6)(3,5,8,4)(10,15)(11,17)(16,22)(18,23)(26,33)(28,35)(32,36)(37,39);(1,4,7,5)(2,8,6,3)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28,31,35)(26,34,33,29)(27,37,30,39)(32,38,36,40);(1,5,7,4)(2,8,6,3)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,34)(26,35,33,28)(27,38,30,40)(32,37,36,39);(1,6,7,2)(3,5,8,4)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30,31,27)(26,36,33,32)(28,39,35,37)(29,38,34,40)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9620580);