#9676641

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	#47542	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,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, 9583837);