#10089194

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	#37780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4392 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9996390);