#8726630

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	#52297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633826);