#9726626

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	#51899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=h,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, 9633822);